Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Brush pork chops with 2 teaspoons white vinegar and sprinkle with 1/4 teaspoon dill, 1/4 teaspoon pickling spice, salt, and black pepper. Place pork chops in a broiling pan.
Broil in the preheated oven until juices run clear and chops are browned, 4 to 8 minutes per side. Drizzle chops with 1 more teaspoon vinegar and flip chops halfway through cooking. An instant-read meat thermometer inserted into the thickest part of a chop should read at least 145 degrees F (65 degrees C).
Place beef broth, potatoes, and 1/4 cup white vinegar in a saucepan, bring to a boil, cover, and boil for 4 minutes. Stir in 1/4 teaspoon dill, 1/4 teaspoon pickling spice, and shredded cabbage; cover and boil for 4 more minutes. Drain liquid and stir 2 tablespoons vinegar into cabbage mixture. Season with salt and black pepper and serve with broiled pork chops.