01 - Set a large cast-iron skillet or griddle over medium-high heat and allow it to get ripping hot.
02 - Divide the ground beef into 4 loosely packed balls, being careful not to overwork the meat.
03 - Place the beef balls onto the hot skillet and smash each one firmly with a metal spatula to form thin patties with jagged, crispy edges. Season the tops generously with salt and pepper.
04 - Let the patties cook undisturbed for about 2 minutes until the edges are deeply browned and crisp. Flip and cook another 1 to 2 minutes on the second side. Remove the patties and set aside.
05 - Reduce the heat to medium. Add the chopped onion, diced bell pepper, and a pinch of salt to the same skillet. Sauté until the vegetables are softened and translucent, about 3 to 4 minutes. Stir in the minced garlic and cook for another 30 seconds until fragrant.
06 - Stir in the ketchup, tomato paste, yellow mustard, smoked paprika, Worcestershire sauce, and chili powder if using. Mix thoroughly to combine all the seasonings with the vegetables.
07 - Pour in the beef broth and bring the mixture to a gentle simmer. Let it cook for about 2 minutes so the flavors meld together.
08 - Return the smashed beef patties to the skillet. Break them up with your spatula and stir them into the sauce. Continue cooking for 2 to 3 minutes until the mixture is slightly thickened and cohesive.
09 - Lay 1 slice of cheese over each mound of the beef mixture. Cover the skillet with a lid briefly, just until the cheese is fully melted and gooey.
10 - While the cheese melts, spread butter on the cut sides of each hamburger bun. Toast them in a separate skillet or on a griddle over medium heat until golden and lightly crisp.
11 - Spoon a generous portion of the cheesy beef mixture onto the bottom half of each toasted bun. Layer on dill pickle chips, shredded lettuce, and tomato slices as desired. Cap with the top bun and serve immediately while hot.