01 - Set a large cast-iron skillet or griddle over medium-high heat and allow it to get thoroughly hot.
02 - Gently 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 press down firmly with a metal spatula to flatten into thin patties with rough, irregular 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 skillet and set aside.
05 - Reduce the heat to medium. Add the chopped onion, 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 an additional 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 let the mixture simmer for about 2 minutes so the flavors meld together.
08 - Return the smashed beef patties to the skillet. Break them up with the spatula and stir into the sauce. Continue cooking for 2 to 3 minutes until the mixture has slightly thickened.
09 - Lay 1 slice of cheese over each mound 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 each hamburger bun. Toast in a separate skillet or on the griddle until golden brown.
11 - Spoon a generous portion of the cheesy beef mixture onto the bottom half of each toasted bun. Top with dill pickle chips, shredded lettuce, and tomato slices as desired. Cap with the top bun and serve immediately while hot.