In a skillet, melt 1 tablespoon of butter over medium heat. Add the sliced mushrooms and onions, then pour in the Worcestershire sauce and red wine vinegar. Cook covered for 10 minutes, stirring occasionally, until the onions soften and turn golden.

Once the onions and mushrooms are caramelized, add the minced garlic. Sauté for an additional 5 minutes until fragrant and softened.

Form four hamburger patties from the ground beef, ensure each patty is roughly ¾ inch thick and season both sides with salt and black pepper. Rub Worcestershire sauce and mustard on each patty.

Set your grill or a skillet to medium-high heat, preheating for a few minutes until hot.

Place the patties on the surface and cook without moving for about 5 minutes. Flip and cook for an additional 3 minutes, then add cheddar cheese on top and cover to melt.

Spread the remaining butter on the insides of the buns and toast until golden brown.

Assemble your burgers by spreading mayo and mustard on buns, adding the cheesy patty, and topping with caramelized onions and mushrooms.