For a sauce combine mayonnaise, Dijon mustard, ketchup, kosher dill pickling brine, a pinch of cayenne pepper in a small bowl. Set aside.
Heat a cast-iron griddle over medium-low heat until warm. Grease the hamburger buns the insides with the melted butter. Place the buns buttered side down on the griddle and toast until golden brown, 2 to 3 minutes. Transfer buns to a plate. Spoon the sauce onto the top bun.
Preheat grill to medium-high. Form the ground beef into 8 round hamburger patties, about ¾ inch thick. Sprinkle with salt and pepper mix, to taste. Grill the burgers, turning once until an instant-read thermometer inserted in the center registers 165°F, 4 to 5 minutes per side. Or you can fry hamburger patties at a cast-iron griddle over medium-low heat up to 4 minutes without moving. Add a slice of cheese on the top of the meat to melt for 1 minute.
Transfer fried ground beef with cheese to the prepared buns. Garnish with lettuce and tomato. Enjoy your meal!!!