Place peppers over an open flame grill or burner, and blacken on all sides. Place in a paper bag, and roll up opening to steam peppers, and loosen charred skin. After 10 minutes, remove peppers from bag, scrape burnt skin away with a paper towel or the backside of a knife. Seed peppers and mince.
Add minced peppers to a large bowl. Add venison, breadcrumbs, salt and pepper. Mix thoroughly with your hands. Form into 4 equal balls. OK, now this might sound a little goofy. Toss the burger balls back and forth into your hands like you would a baseball into a mitt. It will compress the meat and make it hold together better while cooking. Press each ball lightly into a patty, about 1 inch thick.
Combine olive oil and garlic, and brush over one side of the sourdough bread slices.
Grill or broil sourdough bread until lightly browned on the oiled side. Grill or pan-sear burgers over medium heat until cooked to desired doneness. Place cheese on top and cook until melted.
Combine mayonnaise, lemon juice and basil. Place bread on plates, toasted side up. Top with lettuce, tomato, burger and a dollop of mayonnaise mixture.