Make the Big Mac-style sauce first: whisk together the mayonnaise, ketchup, yellow mustard and sweet pickle relish in a small bowl until smooth. Cover and chill while you prepare the beef so the flavors meld.
Divide the ground beef into 8 equal portions and gently roll each into a loose ball — avoid compacting the meat so it smashes easily and stays tender. Sprinkle each ball lightly with salt and pepper, then drizzle Worcestershire over the portioned balls and toss gently to coat.
Heat a large cast-iron skillet or heavy griddle over high heat until very hot. Add the oil and swirl to coat. Working in two batches if needed, place 4 beef balls in the pan spaced apart.
Using a sturdy spatula (or a second spatula and parchment), press each ball down firmly into a thin patty about 3–4 inches wide. Hold the press for 10–12 seconds to get good contact so the edges immediately crisp. Sprinkle a little more salt and pepper over the patties.
Cook the smashed patties without moving them for about 1.5–2 minutes until deeply browned at the edges. Flip each patty, immediately top with a half slice of American cheese (or fold a full slice), and cook another 30–45 seconds. Remove to a plate and loosely tent with foil to rest while you cook the remaining patties.
Warm the tortillas in a dry skillet over medium heat for 20–30 seconds per side or wrap them in foil and heat in a 300°F oven for 5–7 minutes until pliable. Keep them warm wrapped in a kitchen towel.
Assemble the tacos: spread about 1 to 1 1/2 tablespoons of the sauce onto each tortilla, add a handful of shredded lettuce, place one melted smash patty (fold if needed) in the center, then top with diced onion and 2–3 pickle slices. Drizzle a little extra sauce over the top if you like.
Serve immediately so the patties are hot and the cheese is gooey. Leftover sauce keeps well refrigerated for up to 5 days.