Place venison in a freezer for an hour or so until the meat is almost frozen. Using a sharp, thin bladed knife, slice the meat as thinly as possible. Allow to thaw completely, wrap with paper towels to absorb blood, and season liberally with salt and pepper.
Heat olive oil in a large, heavy skillet or griddle over medium-high heat. Add onions, peppers, and garlic. Cook until onions are lightly browned. Add mushrooms, and sauté with onions and peppers until soft.
Move vegetables to one side of the skillet and add sliced venison. Cook meat until lightly brown, but not overcooked. Season with a dash or two of Tabasco.
Mound meat into 4 rectangular piles, about the size of the Italian rolls. Top with equal portions of the vegetable mixture. Top with 2 slices of cheese for each mound, and let melt.
Using two spatulas, scoop up each portion of meat and vegetables, and place in Italian rolls.