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.