Heat 2 tablespoons olive oil in a skillet over medium heat; cook and stir potatoes until browned and crisp, about 15 minutes. Set aside on a paper towel-lined plate.
Place onion and pimiento peppers in the hot skillet; cook and stir until onion has softened, about 5 minutes. Add mortadella and cook until warmed through and slightly browned, 3 to 4 minutes.
Spread each bun with 1 tablespoon spicy mustard. Layer with 2 slices Swiss cheese. Top with 1/4 of the mortadella and onion mixture and sprinkle with 1/4 of the fried potatoes. Top with 2 slices Asiago cheese. Repeat to make 4 sandwiches.
Place sandwiches onto a baking sheet and bake in the preheated oven until cheese is melted and buns are toasted, about 5 minutes.
Heat 1 tablespoon olive oil in a nonstick skillet over medium heat; cook eggs, 2 to 4 at a time if necessary, until whites are set but yolks are still soft, about 2 minutes per side. Place two eggs in each sandwich.