Arrange tostada shells on a baking sheet. Spread a layer of refried beans onto each shell; sprinkle each shell with about 1 tablespoon Cheddar cheese.
Combine chipotle peppers in adobo sauce, salsa, and 1 cup cilantro in a saucepan over medium heat; cook and stir until warm, about 5 minutes.
Put black beans in a microwave-safe bowl; heat in microwave until warm, about 1 minute.
Heat butter in a skillet over medium heat; cook each egg in the melted butter until white is firm and yolk is still together, 2 to 3 minutes. Carefully place each cooked egg atop a tostada shell; sprinkle with remaining Cheddar cheese.
Bake tostadas in the preheated oven until cheese is melted, about 10 minutes.
Place each tostada on a plate; top with salsa mixture and cilantro. Spoon black beans on the side.