01 - Set a large cast-iron skillet or griddle over medium-high heat until thoroughly hot.
02 - Divide the ground beef into 4 loosely packed balls, handling the meat as little as possible to keep it tender.
03 - Place the beef balls onto the hot skillet. Press down firmly with a sturdy metal spatula to flatten into thin patties with jagged, irregular edges. Season the tops generously with salt and pepper.
04 - Let the patties sear undisturbed for about 2 minutes until the edges are deeply browned and crispy. Flip each patty and cook for another 1 to 2 minutes. Remove from the pan 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 30 seconds more 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 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 apart with your spatula and fold the pieces 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 with a lid briefly until the cheese has fully melted.
10 - While the cheese melts, spread butter on the cut sides of each hamburger bun. Toast in a separate skillet or on a griddle until golden brown.
11 - Spoon the cheesy beef mixture generously onto the bottom half of each toasted bun. Add dill pickle chips, shredded lettuce, and sliced tomato as desired. Cap with the top bun and serve immediately while hot.