Foods from the protein group are usually the main ingredient in sandwich fillings, and eggs are no exception. This may include fried eggs, egg salad, or eggs accompanied by other meats, such as bacon, pepperoni, or hamburger; as well as cheeses such as American, mozzarella, or cheddar.
In Cooking Mama, the Egg Sandwich appears as a "Hamburger with Egg". This still counts as an Egg Sandwich, because it's 2 pieces of bread with egg in-between. The burger is just extra.