Whisk together yellow mustard, apple cider vinegar, light brown sugar, hot sauce, Worcestershire sauce, smoked paprika, kosher salt, and black pepper in a bowl until smooth. Taste and adjust balance—add a little more sugar for sweetness, more vinegar for tang, or a pinch more hot sauce for heat. Refrigerate while you prepare the rest; sauce can be made a day ahead for deeper flavor.

Beef Patty

Form the patties

Divide the ground beef into four equal portions (about 4 oz each). Gently form into loose balls and then flatten into thin patties about 4 inches across and slightly larger than the buns. Do not overwork the meat — handle minimally to keep the patties tender.

Season

Season both sides of each patty with kosher salt and freshly ground black pepper just before cooking.

Cook

Heat a cast-iron skillet or griddle over medium-high heat until very hot. Add 1 tsp vegetable oil and spread. Place patties on the hot surface and press down with a spatula for 10–15 seconds to increase contact (optional for a thin smash-style crust). Cook 2–3 minutes on the first side until deeply browned. Flip, immediately place a slice of cheddar on each patty, and cook another 1–2 minutes until cheese is melted and patties reach desired doneness (internal temp ~160°F/71°C for well-done). Remove and rest briefly on a plate.

Bun and Assembly

Toast buns

Split the potato buns and spread the cut sides with softened butter. Toast buttered sides on the skillet or in a pan over medium heat until golden brown and slightly crisp, about 1–2 minutes.

Assemble

Spread a generous tablespoon of Carolina sauce on the bottom bun. Place the cheddar-topped patty on the sauced bun, then add 2 slices of dill pickle, a little thinly sliced red onion, and about 1/2 cup fresh coleslaw. Spoon another 1 teaspoon of Carolina sauce over the slaw if desired. Close with the toasted top bun and serve immediately.

Serve

Serve The Carolina with extra mustard-vinegar sauce on the side and optional kettle chips or fries.