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.