The medieval world burned a lot of firewood, Was there a system for carpenters and woodsmen to save trees perfect for engineering and construction from ending up in a peasant’s hearth?

I can only provide a partial answer to your question based on my research in human landscape ecology. Basically, in many parts of Europe poor forestry management practices were becoming a recognized problem by the later middle ages (~13th-14th centuries). In places like northern France and parts of England (surely elsewhere as well), this was resulting in timber shortages in some areas and a recognized need for sustainability. At this time, the practices of coppicing and pollarding became an increasingly standard practice in woodland areas near human habitation (though it was likely known well before).

Basically, if you've ever observed new shoots of trees sprouting up from a tree's roots or its stump, you're seeing an example of what happens during coppicing. Trees in a forest (if not intentionally planted), were cut down low at the base of the turnk, and new shoots were allowed to grow up from the base. It was then these shoots that would be allowed to age for a about a decade give or take, and be cut, after which new shoots would take their place. Pollarding is a similar practice, but involves cutting branches or trunks higher above the ground (this is what they do to crepe myrtles in the American south today, and you'll see it in parks across Europe) to produce the same effect, while allowing more grassy groundcover that could be used as pasture land or other functions. Done over a large enough area, and you can produce a substantial amount of wood for fuel and poles for light construction for a modest population.

Getting to your question, a coppiced or pollarded tree could live almost indefinitely so long as it was well-managed, providing fuel and light construction material for generations. Meanwhile, among the coppices, select trees could be designated to remain uncut, thus providing larger-scale wood for things like construction. On the specifics of who ordered these practices and how they came to know of them, you'll have to wait for some others to respond as that is out of my depth. I know among feudal laws regarding land use by peasants there were laws on using forest resources (just as you hear the common refrain of "illegally hunting in the king's forest"), but I don't know the specifics around saving large timber wood for construction.

You might want to check out Tibor Hartel and Tobias Plieninger's "European Wood-Pastures in Transition" (2014). I also have "The transformation of traditional woodland management" by Richard Kyser (2009) in my database of sources. It's been a while since I've looked at any of this stuff though, and I was more interested in general woodland management cross-culturally, so wasn't looking too deeply into the specifics of saving construction wood from the peasant's ax.