Loosely shape the beef into 4 equal patties about 1/2-inch thick and 4 inches in diameter. Sprinkle each side of the patties generously with salt and pepper. Create a small well in the center of each patty, using your thumb, to help your burger cook evenly.
Heat a griddle pan over high heat. Add the pastrami and cook until browned and crisp on both sides, 3 to 4 minutes. Transfer to a paper towel-lined plate.
Next, add the burger patties and cook until browned on each side, about 4 minutes per side. Top each burger with 2 slices of the cheese and continue cooking until the cheese is melted.
Slather 1 tablespoon Utah Fry Sauce on each side of the buns. Add the beef patties and top with pastrami and a pile of potato chips. Finish with the top bun and serve with additional chips and sauce on the side.
Mix together the mayonnaise, ketchup, sugar, vinegar and relish in a small bowl. Adjust the seasoning to taste. Refrigerate until ready to use.