Preheat an outdoor grill for high heat and lightly oil the grate.

Melt butter in a skillet over medium-low heat. Cook and stir onion and mushrooms in melted butter until the onions are very tender and beginning to brown, 10 to 15 minutes. Sprinkle brown sugar over the mixture and continue cooking until the onions caramelize, 5 to 10 minutes more; remove from heat and set aside.

Mix ground beef, egg, bread crumbs, horseradish, steak sauce, Worcestershire sauce, garlic, dried onion, salt, and black pepper in a bowl; shape into 4 patties about 6 inches wide and 1/2 inch thick.

Cook burgers on preheated grill until completely browned, about 3 minutes per side. Reduce heat to medium and cook burgers another 3 minutes per side, topping each burger with a slice of pepperjack cheese for the last minute of cooking.

Top each burger with about 1/4 of the mushroom and onion mixture, 3 pieces of bacon, and 1 tablespoon guacamole to serve.