Are you possibly thinking of Richard Lionheart? He was trying to return home after the Third Crusade, but due to the various people he had offended during the crusade, he couldn't pass through France or the Holy Roman Empire. The only part of the Empire he could safely visit was the Duchy of Bavaria where his sister was married to the duke. Unfortunately for Richard he got caught outside Vienna, in the Duchy of Austria.

There are various stories about how he was caught, but he may have been dressed as a pilgrim, or perhaps as a cook - supposedly, when Leopold’s men found him he was cooking a chicken. He was discovered either because he (or his men) had been spending a suspiciously large amount of money, or because he was wearing a ring that was obviously too valuable for the type of person he was pretending to be. The French and German sources that are hostile to Richard tried to portray him as foolishly as possible, and English sources either portray him more heroically or avoid giving specifics about his capture, so it’s hard to know exactly what happened.

I wrote more about Richard's capture in a previous answer. I haven't heard a "12-egg omelette" version of this story though, so maybe you're thinking of someone else entirely...

There is a story about Charles II fleeing after the battle of Worcester in 1651 that has a reference to eggs, but it is possibly a coincidence with your question. The story (and I call it a story) was partially told by Charles thirty years after the event and recorded by Samuel Pepys, and the other half comes from a 1660 pamphlet detailing Charles' escape from England (An Exact Narrative and Relation of His Most Sacred Majesties Escape from Worcester, Wing E3662).

After the Royalist army was defeated in Worcester, Charles fled towards Wales. His greatest problem was that he was constantly recognized wherever he traveled. As Ronald Hutton wrote, "He was constantly recognized by former royalist soldiers, sailors, or courtiers. Apart from the butler at Abbots Leigh, such individuals included a vagrant on the road in Dorset, an ostler at Bridport, a gentlewoman in Wiltshire, an innkeeper at Brighton, and the master of the ship on which he crossed the channel."

On September 4, 1651, he arrived at a house called "Whiteladies", owned by a recusant family, the Giffords. There he was disguised as a servant, hastily cut his hair with a knife, and (as the story goes) rubbed soot from the chimney on his face. He was forced to hide in the woods near the house the same night, during a downpour. While hiding, a royalist who had helped guide Charles from the battle to Whiteladies, Francis Yates, sent his sister into the woods with food for Charles - scrambled eggs. The pamphlet reads "a Messe of Milk, Eggs and Sugar in a black earthen Cup, which the King...and said, he loved it very well." Another source states that Charles was concerned about a woman knowing his whereabouts, and relays the following exchange:

Charles: Good woman, can you be faithful to a distressed Cavalier?
Yates: Yes Sir. I will die rather than discover you.

There is nothing about a 12 egg omelet in the story, but it does have the elements of an English King fleeing, disguising himself, being regularly recognized, and enjoying a meal of eggs during his flight.

Sources:

Hutton, Charles the Second
An Exact Narrative and Relation of His Most Sacred Majesties Escape from Worcester, Wing E3662
Ollard, The Escape of Charles II after the Battle of Worcester
"An Account of His Majesty's Escape from from Worcester", printed in Hamilton and Scott, Memoirs of the Court of Charles II

Another historical personality with whom this story is connected is a French Enlightenment figure, the Marquis de Condorcet. During the French Revolution, he attempted to escape the Revolutionary authorities in disguise, but was captured—perhaps because of ordering a jumbo omelette-- and died in jail under unclear circumstances.

You can easily do an online search using his name and find many retellings of the omelette story. Here’s one version, from the 11th edition of the Britannica (1910):

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On the evening of the 13th of April 1795 – not, as Carlyle says, on a “bleared May morning,” –with garments torn, with wounded leg, with famished looks, he entered a tavern in the village named [the village of Clamart], and called for an omelette. “How many eggs in your omelette?” “A dozen.” “What is your trade?” “A carpenter.” “Carpenters have not hands like these, and do not ask for a dozen eggs in an omelette.”

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There’s an 1897 biography of Condorcet, by J. F. E. Robinet. His account of Condorcet’s arrest doesn’t mention omelettes, as far as I could see. But in an appendix he does include this, written by Condorcet’s daughter: “It is certain that hunger drove him to go to an inn, where the large number of eggs in an omelette that he ordered attracted attention.” Perhaps an authority on Revolutionary France will weigh in!

Just as an addition to others posts. There is a legend in Faversham that James II, after fleeing from William of Orange, was staying at an inn in the town and the innkeeper recognised him and arrested him. I believe its a shoe shop now.

As far as I am aware the story isn't accurate and the accepted account is that he was captured by some local fishermen during a storm and taken to the Queens Arms pub in the market. There is a brief account of it in Archaeologica Cantiana which you can find online for free. There is also a brief mention of it in the Oxford Dictionary of National Biography but that would need an institutional log in for access.

Sydney Wilson, 1932, 'King James II at Faversham', Archaeologia Cantiana, 44:269-270.

https://kentarchaeology.org.uk/node/10655

Ward, A. James II (1633–1701). Oxford Dictionary of National Biography. https://www.oxforddnb.com/view/10.1093/odnb/9780192683120.001.0001/odnb-9780192683120-e-14593.