Melt the butter in a skillet over medium heat. Add the thinly sliced onion and cook low and slow, stirring occasionally, for 15 to 20 minutes until soft, deeply golden, and slightly jammy. Remove from heat and let cool slightly.

In a large bowl, combine the ground beef, ground pork, egg, all the seasonings, Worcestershire sauce, and 1/4 cup of breadcrumbs. Mix gently until just combined. Do not overwork the mixture.

Fold in the cooled caramelized onions. If the mixture feels too soft to hold together, add more breadcrumbs one tablespoon at a time until it holds a patty shape without falling apart.

Divide into 6 to 8 equal portions and gently shape into thick patties. Press a small dent in the center of each patty with your thumb to prevent them from puffing up during cooking.

Preheat the grill to medium-high heat. Grill the patties for 4 to 5 minutes per side depending on thickness until a beautiful char forms on the outside and the burgers are cooked through and juicy.

Add cheese in the last minute of cooking and close the grill lid to melt. Serve on toasted buns with extra caramelized onions and your preferred toppings.