Melt the butter with the olive oil in a large skillet over medium heat. Add the sliced onions and salt, cooking for about 25 to 30 minutes. Stir occasionally and lower heat as they soften.

During the last 2 minutes of cooking, stir in the balsamic vinegar. Remove from heat and divide the mixture for stuffing and topping.

In a bowl, gently mix ground beef with kosher salt, black pepper, and Worcestershire sauce without overworking. Divide into 8 equal portions and flatten into patties.

Spoon tablespoon of cooled caramelized onions onto the center of 4 patties, then place the remaining 4 patties on top and press edges to seal.

Make a small thumbprint in the center of each patty.

Heat a skillet or grill pan over medium-high heat. Cook burgers for 4 to 5 minutes on each side until they reach a safe 160°F internal temperature.

During the last minute, place cheese on each burger and cover briefly to melt.

Spread mayonnaise on the bottom of toasted buns. Add lettuce, tomato slice, stuffed burger patty, and caramelized onions. Close with bun tops and serve immediately with pickles.