I believe you are referring to the Soviet concept of "Deep Battle."

For a bit of context I'm going to quote historian David Glantz on the matter:

Perhaps most important, the brilliant Civil War commander M. N. Tukhachevsky and the military theorist V. K. Triandafillov developed a strategic theory of successive operations based on the Soviet military failure against Poland in 1920 and the failed German offensives against France in

1918. Put simply, they believed that modern armies were too large and resilient to be defeated in one cataclysmic battle. Instead, ,the attacker would have to fight a series of offensive battles, each followed by a rapid exploitation into the enemy rear and then another battle when the defender reorganized his forces.6 To place these battles in a common strategic context, Soviet soldiers began to think of a new level of warfare, midway between the tactics of individual battles and the strategy of an entire war. This intermediate level became known as Operational Art (operativnaia iskusstva).

Operational Art may be thought of as the realm of senior commanders who plan and coordinate operations of large formations within the context of a strategic operation or an entire campaign, that is, a series of actions culminating in the achievement of a strategic objective. In 1927, Svechin summarized this theoretical structure: "Tactics make the steps from which operational leaps are assembled, strategy points out the path."7

Basically, Deep Battle was the idea that rather than breakthrough in one area, you would use overwhelming numerical superiority to breakthrough in multiple areas, preventing the enemy from effectively forming a defensive line. A Soviet officer named I.A. Egorov put it as such:

The principal and basic task of military art is to prevent the formation of a firm front [by the defending enemy], imparting a destructive striking force and a rapid tempo to operations."

Did the hierarchic system have to change for it in the lower or even higher ranks? Were different vehicles and weapons needed?

In terms of technology, when the theory was developed it was intended to be accomplished with the weapons used in the Russian Civil War. So artillery, cavalry, Infantry, and armored cars. However, as technology moved forward, Deep Battle was expanded to include the use of Tanks, air planes, and camouflage. So new technology made Deep Battle more effective, but it wasn't necessarily crucial for the plan to work.

As for a new hierarchic system the same principle applies. Deep Battle works best when you have independent tank armies. The Germans used their independent Panzer armies to great effect in the war. The Soviets didn't have an equivalent at the start of the war; the closes they had were called "Mechanized corps," again quoting Glantz:

Soviet armored formation in 1941 was the mechanized corps, a rigid structure that contrasted unfavorably with the easily reorganized German motorized corps. Each mechanized corps was centered around two tank divisions, each authorized 10,940 men and 375 tanks organized in two tank regiments, one motorized infantry regiment, and battalions of reconnaissance, antitank, antiaircraft, engineer, and signal troops. Such divisions were imbalanced, with far more tanks than other combat arms elements, although, to compensate for this, mechanized corps also included a motorized infantry division and various support units, for an authorized total of 36,080 men.

Eventually this was remedied with the creation of "Shock armies."

How could the Nazis have countered this doctrine? Not like in "could they have won?" but more like "How could they do more significant damage?"

The counter to Deep Battle is something called "Elastic Defence." Basically you retain a strong mobile striking force to deal with breakthroughs or to use as reserves and more importantly you allow your men to withdraw temporarily if they are under heavy pressure. Von Manstein was a proponent of this. But Hitler didn't like that idea, and often times viewed retreat, even temporary withdraws as cowardly. Hitler preferred to use the defensive strategy Germany had used in WW1 (with regards to defence, Hitler wasn't a luddite and was open to new ideas, especially early in the war). The creation of massive fortifications with mass infantry in defence. The idea of fortresses to slow the enemy down and hopefully provide bases for an eventual counterattack was well founded in German doctrine circa late WWI. This failed because A) Germany didn't have the necessary amounts of barbed wire, mines, etc. to properly fortify and B) because the Germans often didn't have the infantry manpower reserves to properly fight these types of defensive battles. Hitler's insistence on German infantry maintaining their positions often put them at huge risk to massive Soviet artillery bombardments, further weakening the German position.

Is there something interesting to know about the operations per se? Where did a textbook example of this take place?

The main proponent of the theory, Mikhail Tukhachevsky, had been killed as part of Stalin's purges. As such when Germany invaded the Soviets struggled to properly implement the concepts of Deep Battle (also not helped by the lack of independent tank armies). However, by the end of 1941 the Soviets were recovering and began to make use of Shock armies on the attack. This culminated in the textbook example of "Deep Battle" which was operation Bagration.

Bagration took place in June of 1944. The offensive was planned to liberate Belarus, and hopefully destroy German Army Group Centre. To start the Soviets launched a massive deception campaign (one of the key aspects of Deep Battle) trying to convince the Germans that the offensive would be launched North and South of Belarus. It also helped that the Germans were more concerned with Allied landings in France and the German air force was far too occupied defending mainland Germany from bombers. German units in Belarus were also thinned out trying to hold such a large area with very few reserves.

The actual battle started the night of June 22nd and was followed by many different offensives.

In the Vitbesk sector Third Panzer army was hit first by two Soviet armies, a German army corps was trapped in the city of Vitbesk and Third Panzer army was mauled and the remnants began to flee back to Minsk.

The German fourth army was hit next by the 2nd and 3rd Belorussian fronts and the city of Mogilev was taken by June 28th. The German ninth army was next up on the chopping block. On June 24th it was hit by the Soviet 1st Belorussian front, in many places the Germans simply dissolved under the weight of the attack. Eventually the entire German ninth army was encircled.

All along the German front, Soviet armies penetrated deep into thinly held German positions and then encircled the retreating German units. Minsk was taken on July 3rd, just over a week after the offensive.

The end result of this was that Army Group Centre was almost completely wiped out, and Germany was now pushed back into Poland where the Soviet offensives stalled. Bagration was a huge success and perfectly showed what Deep Battle could do under ideal circumstances.

Edit:

Wording

Also sources:

David Glantz When Titans Clashed where I got the quotes from

General other works on the Eastern Front/WWII ask if you want any other recommendations but the Glantz book is a good starting point.

How could the Nazis have countered this doctrine? Not like in "could they have won?" but more like "How could they do more significant damage?"

The methods used to achieve breakthrough were kinda different for the major armies but generally in the grand sheme of things the strategy was the same. You mass your troops and break through the enemy defense line and then start to manouver into the open while destroying supply lines commando chains et cetera and eventually destroy the troops which were holding the frontline at different areas. Some had more focus on air support some attacked on narrow fronts like the Western allies, the Red Army preferred attacks on several sectors. How do you stop that?

In theory this is easy: you have mobile troops which are positioned behind the line and counterattack either directly the enemy spearheads or, if the force ratios allow for it and the enemy overextended, the flanks of the enemy movement and cut off the spearhead. This would likely be done with strategic reserves. Its important to find the right ratio of frontline forces to reserves. The frontline has to be strong enough to inflict major damage to the attack and slow down the advance but if too many forces are at the front the units avaible as reserve are to weak to counterattack.

While the Eastern Front overall was obviously a decisive victory for the Soviet Union the attempts to break through German lines more often than not failed and cost alot of casualties. It was more the weight of the Red Army what pushed the Wehrmacht back than decisive breakthroughs. The most notable success was for most historians Operation Bagration which saw the destruction of Army Group Mid ( Heeresgruppe Mitte ) this operation was remarkable for its swiftness and overall casualty ratios. While the Red Army lost more soldiers than the Wehrmacht the numbers were far less loopsided than before. The huge amount of prisoners indicated that the Wehrmacht was not just overwhelmed by numbers but overrun and outmanoeuvered.

But

The Red Army even from the beginning tried to encirle major Wehrmacht forces instead of approaching a slower strategy, this often resulted in serve defeats. A good example for such defeat is the Second Battle of Kharkov, major Soviet forces were destroyed after heavy counterattacks encirlced several armies. There are many examples for such counterattacks which in return encircled and destroyed attacking armies. Operation Mars highlights the general issue with Soviet offensive Operation against German troops, while in theory the "Deep Battle" doctrine was fine it was similiar to the Wehrmacht concepts, the tactical requirements were not met in the early phase of the war. The force ratios were not enough and the involved troops to far behind in terms of quality. The Soviet leadership didn't recognize this and lost maybe more men than with a skilled defense. During Operation Mars which was a similiar operation to Uranus ( Stalingrad ) the Soviet attack run into German forces in contrast to Uranus which met Italian and Romanian forces. Several spearheads which intented to cut of the 9th Army were in return cut of themselfs and anhilated resulting in a sharp defeat with a 8:1 casualty ratio ( Glantz).

The choice of the methods/tactic is obviously influenced by the operational situation, force ratios, force quality, terrain et cetera. In Normandy for example the Wehrmacht would often put low grade divisions in the frontline and pool more powerful forces behind the frontline. The low grade divisions would take the hits of the first attacks and later if tactical breakthroughs were achieved the mobile forces would attack the spearhead and often destroy them. The operations around Cean show this perfectly. The forces present were not enough for major counterattacks so the tactic of choice was to attack the spearheads and contain the offensive ( Stabilizing or reestablishing the front and regain strongpoints like hills et cetera ). If the situation becomes critical and a breakthrough is too deep a major counter attack is often tried. Operation Lüttich is here the perfect example, the gap in the German lines was huge and so it was decided to cut of the entire advancing forces with a "major" counterattack. Since no real strategic reserves were available units were shuffeled around to create battlegroups. The operation failed but highlights the general approach to counterattacks. Another option is retreat which isn't neccessarily a bad choice. The Wehrmacht in general was very open to taking risks while the Western Allies especially the Commonwealth forces under Montgomery were cautios likely influenced by the different strategic situations. The Battle of the Bulge is a good example for that. The penetration was pretty deep but in a relative thin corridor opening the attacking forces up to flanking but it was decided to push the enemy back and restore the frontlines. General Patton is the best known proponent for more aggressive risky movements.

Is there something interesting to know about the operations per se? Where did a textbook example of this take place?

Like mentioned earlier most historians agree that the Operation Bagration was the biggest defeat of the Wehrmacht.

"How could they do more significant damage?"

The Wehrmacht often choose not to retreat when retreat was the best choice and implemented hedgehog defense which resulted in unneccessary casualties. Some of these decisions were made by Hitler who generally was very opposed to retreat. The encirclements, Normandy et cetera were all forseen by pretty much every high ranking officer and orderly retreat with local counterattack would have certainly inflicted more damage but you can't retreat forever obviously.