01 - Set a large cast-iron skillet or griddle over medium-high heat and allow it to get thoroughly 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. Press down firmly with a metal spatula to flatten into thin patties with rough, jagged edges. Season the top side with salt and pepper.
04 - Let the patties sear undisturbed for about 2 minutes until the edges are deeply browned and crispy. Flip and cook for another 1 to 2 minutes. Remove the patties from the pan and set aside.
05 - Lower the heat to medium. Add the chopped onion, diced bell pepper, and a pinch of salt to the same skillet. Sauté until the vegetables have softened, about 3 to 4 minutes. Stir in the minced garlic and cook for 30 seconds more until fragrant.
06 - Add the ketchup, tomato paste, yellow mustard, smoked paprika, Worcestershire sauce, and chili powder if using. Stir everything together until well combined.
07 - Pour in the beef broth and let the mixture simmer for about 2 minutes so the flavors meld together.
08 - Return the smashed patties to the pan. Break them apart with the spatula and fold them into the sauce. Continue cooking for 2 to 3 minutes until the mixture has slightly thickened.
09 - Lay 1 slice of cheese over each portion of the beef mixture. Cover the skillet briefly to allow the cheese to melt evenly.
10 - While the cheese melts, spread butter on the cut sides of the hamburger buns and toast them in a separate skillet or on the griddle until golden brown.
11 - Spoon the cheesy beef mixture onto the bottom half of each toasted bun. Top with dill pickle chips, shredded lettuce, and sliced tomato as desired. Place the top bun and serve immediately while hot.