Preheat the oven to 350 degrees F (175 degrees C).
Mix beef, mayonnaise, and green pepper together in a large bowl. Form into 12 palm-sized balls and dip into beaten egg to coat. Roll in crushed crackers and flatten into a patty.
Heat a skillet over medium heat. Fry patties in batches until browned, about 3 minutes per side. Place patties into two 9x13-inch baking pans.
Cover each patty with a slice of cheese. Mix each can of soup with 1/3 cup milk and pour over the patties.
Bake in the preheated oven until cheese is melted and soup starts to bubble, 20 to 25 minutes.
