Mix ground beef, egg, 1 Tbsp Worcestershire sauce, ¼ tsp salt and ¼ tsp pepper in a large bowl. Form patties that are 4 inches in diameter and ¾ inch thick.
Grill the burger patties on a grill preheated to 375–400 ºF. Cook for 10 minutes on medium heat. Flip the patties and cook for 5 additional minutes. Place 1 slice of Swiss cheese on each burger and cook for 1 minute until the cheese melts. Meanwhile, toast the brioche buns on the grill for 1 minute. Remove the burger patties and buns from the grill.
While the burgers are grilling, slice the onions into ¼ inch thick slices and mince the garlic. In a large pan, cook the butter and minced garlic for 1 minute over medium heat. Add the onion slices and cook for 5 minutes. Turn the heat up to high and add mushroom slices, 2 Tbsp Worcestershire sauce, ¼ tsp salt, and ¼ tsp pepper. Cook for 10 minutes while stirring until the onions and the mushrooms are browned. Remove from heat.
Assemble each burger by placing the bottom bun, the burger patty, ½ cup of mushroom and onion topping, and lastly the top bun.