Melt the butter in a large frying pan over medium heat. Add in the onions and stir to coat the onions with the butter. Cook for 45 minutes to an hour, stirring often, and adjusting the heat as needed, until the onions are deep brown and reduced in size. Add in the thyme and Worcestershire and cook an additional 2 minutes. Remove from heat.
Preheat another large nonstick or cast-iron pan over medium heat with a little vegetable or peanut oil. Add in the burgers and cook about 5 minutes until you see the juices coming to the top of the burger patties. Flip and cook an additional 5 minutes until cooked through.
Preheat broiler to high. Put the bread on a baking sheet and broil to lightly toast the bread on one side. Remove from the broiler. While your bread is broiling, you can grate your cheese in preparation for the next step.
Flip the bread over, add the burger patties to the same pan. Top all the bread and burgers with cooked onions, then top the onions with grated cheese. Return the pan to the broiler and cook to melt and brown the cheese on top.
Build the burgers with one piece of bread on the bottom, then the burger, then flip over another piece of bread and put it on top so the cheese side is down. Serve.