Caramelize the Onions: In a large skillet over medium heat, melt the butter. Add the sliced onions and sugar, and cook, stirring occasionally, until the onions are deeply caramelized and sweet, about 20-25 minutes. Add a splash of beef broth if the onions start to stick, to deglaze the pan.Prepare the Burger Patties: Season the ground beef with salt, pepper, and Worcestershire sauce. Form into 4 equal-sized patties.In a separate skillet or on a grill, cook the burger patties to your desired level of doneness.Assemble the Burgers: If desired, prepare garlic butter buns by spreading buns with a mixture of butter and minced garlic, then toasting until golden.On each bun bottom, place a cooked burger patty. Top each patty with a generous helping of caramelized onions and a slice of cheese.Broil the assembled burgers for a few minutes until the cheese is melted and bubbly.Sprinkle fresh thyme over the cheese, cover with bun tops, and serve immediately.