Preheat a cast-iron skillet or griddle pan over medium-high heat. Combine the chuck Worcestershire, onion, salt and lots of pepper. Form into 4 patties (thinner at the center and thicker at the edges for even cooking). Add oil to the hot pan, add the patties and cook, turning once for 8-10 minutes for medium-rare to medium. Top the patties with the cheese, folded to fit the burgers, during the last 2 minutes of cooking.
Meanwhile, in a separate skillet, melt the butter over medium heat. Add the mushrooms and cook until browned, about 12 minutes. Add the shallot, season with salt and pepper and cook for 2-3 minutes more. Deglaze the pan with the sherry and remove from heat.