Preheat the oven to 350 degrees F (175 degrees C).
Heat oil in a frying pan over medium-high heat. Season meat with salt and pepper and coat in flour. Brown meat on both sides, 3 to 5 minutes, and transfer to a baking dish.
Melt butter in the same frying pan and cook mushrooms until tender, scraping the bottom of the pan, 5 to 7 minutes. Season with salt and pepper. Add mushroom soup and sherry; stir gravy until well incorporated.
Pour gravy over the meat in the baking dish. Cover with aluminum foil.
Bake in the preheated oven until meat is no longer pink in the center and an instant-read thermometer inserted into the center reads 145 degrees F (63 degrees C), about 25 minutes. Remove aluminum foil, stir gravy, and top burgers with mozzarella cheese.
Return to the hot oven and bake until cheese is melted, 3 to 5 minutes more.
Serve with Garlic Mash potatoes, creamed spinach and Texas toast