In a large skillet over medium heat, cook the ground beef, onion, and garlic until the beef is browned and the onion is soft. Drain any excess fat.

Stir in the ketchup, yellow mustard, Worcestershire sauce, apple cider vinegar, and brown sugar. Season with salt and pepper to taste. Let it simmer for 5 minutes until the mixture thickens.

While the beef simmers, prepare the special sauce by mixing mayonnaise, French dressing, sweet pickle relish, minced onion, white vinegar, sugar, and salt in a small bowl until well combined.

Toast the hamburger buns lightly if desired. Spread a generous amount of special sauce on the bottom buns.

Spoon the beef mixture onto the buns, then top with shredded lettuce, pickles, and shredded cheddar cheese.

Place the top buns on the sandwiches and serve immediately.