Combine mustard, ketchup, and cherry peppers in the bowl of a food processor. Pulse until smooth and set aside.
Melt butter in a small skillet over medium-high heat. Add onions and cook until onions are soft and are starting to caramelize, about 5 minutes. Remove from heat and set aside.
Form beef into two patties and season with salt and pepper. Heat a large cast iron skillet over medium-high heat. Add hamburger patties and cook 5 minutes or until a crust forms on the bottom. Flip patties over and cook 3 minutes.
Using a wide spatula, break patties up into large pieces. Sprinkle sautéed onion in between spaces of the hamburger pieces. Layer cheese over beef, and cook 2 minutes or until cheese is melted.
Spread reserved mustard sauce on bottom bun of each hoagie roll. Divide meat mixture between two sandwiches. Top each with shredded lettuce and sliced tomato.