In a small bowl, stir together mayonnaise, ketchup, pickles or relish, sugar, and vinegar. Set aside or refrigerate if using later.

Portion the ground beef into four 5-ounce balls, pressing together just until the meat holds its shape without falling apart.

Add oil to a skillet and heat until smoking. Add one of the meatballs to the skillet and press down with a spatula until it's about the diameter of the tortilla. Season with salt and pepper.

Cook undisturbed until a dark brown crust forms, about 2-3 minutes. Carefully flip the patty over, scraping up browned bits.

Top patty with 2 cheese slices, cook for 30 seconds, then add pickles and onion. Add a tortilla and press down firmly.

Flip the patty again, adding 2 more slices of cheese, then place the second tortilla on top. Press to adhere and cook briefly until the cheese melts, about 30 seconds.

Remove from the pan and repeat with remaining balls of meat, tortillas, and toppings.

Slice each quesadilla into 6 equal triangles, top with shredded lettuce if using, and drizzle with special sauce or serve it on the side.