Almost-Famous Animal-Style Burgers

INGREDIENTS

• Freshly ground pepper
• Kosher salt
• 1/4 cup mayonnaise
• 2 tablespoons ketchup
• 1/4 cup sliced dill pickles
• 2 large onions, finely chopped
• 4 hamburger buns, split
• 1/2 teaspoon white vinegar
• 8 slices American cheese
• 2 pounds ground beef chuck (preferably 60 percent lean)
• 1/4 cup yellow mustard
• 4 to 8 thin slices tomato
• 1 tablespoon sweet pickle relish
• 3/4 cup shredded iceberg lettuce
• 2 tablespoons vegetable oil, plus more for brushing