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.

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

Flip the patty over again, adding more cheese and placing the second tortilla on top. Press to adhere.

Flip once more and cook briefly to allow the extra cheese to melt, about 30 seconds. Remove from the pan and repeat.

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