Make the Big Mac Sauce first — Whisk together the mayonnaise, French dressing, sweet pickle relish, white wine vinegar, onion powder, garlic powder, and paprika in a small bowl. Pop it in the fridge while you cook the beef. The flavors get even better as it sits.
Brown the beef — Heat a large skillet over medium-high heat. Add the ground beef and cook, breaking it apart, until no pink remains, about 6–8 minutes. Drain off any excess fat, leaving just a little behind for flavor.
Soften the aromatics — Reduce heat to medium and add the diced onion. Cook for 3–4 minutes until softened, then stir in the minced garlic and cook for another 30 seconds.
Build the sloppy joe base — Stir in the ketchup, yellow mustard, white wine vinegar, Worcestershire sauce, onion powder, garlic powder, and paprika. Mix everything together well and let it simmer on low for 5–7 minutes until the sauce thickens slightly. Season with salt and pepper.
Toast the buns — Lightly butter the inside of your sesame seed buns and toast them in a skillet or under the broiler until golden. This step is worth the extra two minutes — it keeps the buns from getting soggy.
Assemble — Spoon the meat mixture onto the bottom bun, lay a slice of American cheese on top (it'll melt from the heat of the meat), drizzle generously with Big Mac sauce, then pile on shredded lettuce, pickle slices, and minced raw onion. Cap it off with the top bun and serve immediately.