Heat butter and olive oil in a skillet over medium-low heat. Add onions and a pinch of salt. Cook slowly for 30-45 minutes, stirring occasionally. Deglaze with broth if needed.

Once onions are deep brown and jammy, stir in garlic and thyme. Cook 1 minute, then set aside.

Divide ground beef into 4 equal portions and form into patties. Create a shallow dimple in the center of each. Season generously with salt and pepper.

Lightly butter or mayo the cut sides of the brioche buns and toast them face-down in a skillet until golden brown. Set aside.

Heat a cast-iron skillet over medium-high heat. Add patties and cook 3-4 minutes on the first side until a crust forms.

Flip patties. Immediately top with warm caramelized onions and a slice of cheese. Cover the pan to melt the cheese and cook for 3-4 more minutes.

Assemble the burgers by placing the patties on the toasted buns, garnishing with fresh thyme, and serving immediately.