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 12-inch skillet and heat until smoking. Add one of the meatballs to the skillet and press down until it is roughly 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 all the browned bits.

Top patty with 2 cheese slices and cook for 30 seconds, then add 5-6 pickles and 1-2 tablespoons of onion. Add a tortilla and press down firmly to make sure the cheese adheres.

Flip the patty over again, adding 2 more slices of cheese, and place the second tortilla on top. Press to adhere. Flip once more and cook briefly, about 30 seconds. Remove from the pan and repeat the process with the remaining ingredients.

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