In advance, prep the onions. French 2 onions and cook in olive oil and butter over medium heat stirring often until deeply caramelized, about 30-40 minutes. Refrigerate.
Form sausage into 6 equal-sized patties. Start cooking sausage in the pan over medium heat.
Prep the croissant. Cut in half and spread with a light coating of dijon mustard and and even lighter amount of ketchup.
When the patty is flipped, add onions to the pan to begin reheating.
When sausage is done, put the onions on top of the sausage patty then top with cheese to melt. The cheese will keep the onions from sliding around too much when the sandwich is bitten into.
While the cheese melts, crack the egg into the pan, lightly scramble, and add salt and pepper. When desired doneness is reached, put egg on top of cheese.
Transfer assembled stack to prepared croissant, cut in half, and serve.