01 - Place a large cast-iron skillet or griddle over medium-high heat and allow it to get fully hot before adding any ingredients.
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 create thin patties with rough, jagged edges. Season the tops with salt and pepper. Let the patties cook undisturbed for about 2 minutes until the edges are crispy and deeply browned. Flip and cook for another 1 to 2 minutes. Remove the patties and set aside.
04 - Reduce the heat to medium. Add the chopped onion, bell pepper, and a pinch of salt to the same pan. Sauté until the vegetables are softened, about 3 to 4 minutes. Add the minced garlic and cook for another 30 seconds until fragrant.
05 - Stir in the ketchup, tomato paste, yellow mustard, smoked paprika, Worcestershire sauce, and chili powder if using. Mix everything thoroughly to combine.
06 - Pour in the beef broth and bring to a simmer for 2 minutes, allowing the flavors to come together.
07 - Return the smashed beef patties to the pan, breaking them up with the spatula and folding them into the sauce. Continue cooking until the mixture is slightly thickened, about 2 to 3 minutes.
08 - Place 1 slice of cheese over each mound of the beef mixture. Cover the pan briefly to allow the cheese to melt evenly.
09 - While the cheese melts, spread butter on the cut sides of the hamburger buns and toast them in a separate skillet or on a griddle until golden brown.
10 - 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. Cap with the top bun and serve immediately while hot.