Add 1 tablespoon of the butter, chopped onions and bell pepper to an 8-quart saucepan over medium heat. Cook, stirring occasionally, until the onions are softened, 2 to 3 minutes. Add the tomato paste and cook, stirring, until the tomato paste is toasted, 2 to 3 minutes. Add the garlic and crushed tomatoes and stir well. Add the hot sauce, lemon juice and seafood boil seasoning. Cook until the vegetables are tender, 4 to 5 minutes. Scoop up the onions and bell peppers with a slotted spoon and keep warm. Let the sauce cool.
Meanwhile, heat a large cast-iron skillet over high heat. Season the beef with salt and pepper. Divide into 4 equal patties and transfer to the skillet. Cook for 4 to 5 minutes. Add the remaining 1 tablespoon butter and flip the patties. Top each patty with 2 slices of cheese and bake just until the cheese is melted, 2 to 3 minutes.
Meanwhile, heat a cast-iron griddle over medium heat until hot. Gently toast the buns split-sides down just until they brown slightly, 1 to 2 minutes. Set aside.
Combine the mayonnaise and up to 2 tablespoons of the cooled sauce in a small bowl and stir well. Slather 1/2 tablespoon on each of the split sides of the buns.
Spoon the onion and peppers onto the bottom buns then top each with a burger patty.
Top each burger with 1 tablespoon of the fried onions. Close the burgers and serve hot.