Preheat a grill to medium-high heat.
Divide the ground beef into quarters and form into even patties. Gently flatten the patties to 1/2-inch thick and press your thumb into the center to make a slight divot. Place a rounded tablespoon of the grated cheese into each divot. Fold the edges of the meat over the cheese to completely envelop and press to make a nice seal. Reshape and flatten the patties, if needed.
Sprinkle the patties with salt and pepper. Grill until the meat is browned more than halfway up the bottom, about 5 minutes, and then flip and cook until the bottom is nicely browned with grill marks, another 4 minutes for medium rare, or another 6 minutes for medium.
Heat the oil in a large saute pan and add the mushrooms in a single layer. Saute until browned and crispy on the bottom, about 2 minutes, and then flip. (Don't worry if the mushrooms curl up; flipping will flatten them out.) Cook for another 3 minutes, and then remove from the heat.
To assemble the burgers, spread 1/2 tablespoon mayonnaise on the bottom of each roll and divide the arugula on top. Place the burgers on the arugula, scatter over the mushrooms and top with the remaining buns.
From Vince Camillo for Cooking Channel