1) Preheat oven to 420 degrees. Generously coat both sides of steak with salt and pepper. Heat 1 Tbsp. butter on medium high heat on skillet. Sear both sides of steak, about 4 minutes per side. Place steak on baking pan and bake for 20 minutes, flipping over once after 10 minutes. Take out of oven, let rest for about 5 minutes before cutting against the grain in thin slices.
2) Once the steak is in the oven, use the same skillet for sautéing peppers and onions. Sauté until onions are translucent and caramelized. Turn heat off and set peppers and onions aside.
3) Combine mayo and horseradish together, add salt and pepper to taste. Spread on one side of each hamburger bun. Add sliced steak, cheese, and top with peppers and onions. Serve.