In a large skillet, cook the ground beef, onion, and garlic over medium heat until the beef is browned and the onions are soft. Drain any excess fat.
Stir in ketchup, mustard, Worcestershire sauce, apple cider vinegar, and brown sugar into the beef mixture. Season with salt and pepper to taste. Simmer for about 5 minutes, or until the mixture has thickened slightly.
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 and start assembling the Sloppy Joes: spread a generous amount of special sauce on the bottom buns, then top with the beef mixture, shredded lettuce, pickles, and cheddar cheese.
Cap with the top buns and serve immediately.
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