01 - In a small bowl, combine mayonnaise, ketchup, yellow mustard, sweet pickle relish, garlic powder, smoked paprika, white vinegar, and a pinch of black pepper. Whisk thoroughly. Cover and chill until needed.
02 - Heat a large skillet over medium-high heat. Add the ground beef and cook, breaking it apart with a spatula, until browned and thoroughly cooked, approximately 5-6 minutes. Drain any excess fat. Stir in kosher salt, black pepper, onion powder, garlic powder, and ketchup. Mix well to combine, then remove from heat.
03 - Wipe out the skillet and reduce heat to medium. Lay one flour tortilla flat. Sprinkle one quarter of the cheddar and mozzarella cheeses evenly over half of the tortilla. Top with one quarter of the cooked beef mixture, diced dill pickles, and diced red onion. Drizzle a spoonful of the prepared special sauce over the filling.
04 - Fold the filled half of the tortilla over the filling to create a half-moon shape.
05 - Melt a portion of the unsalted butter in the skillet. Carefully place the folded quesadilla in the skillet and cook for 2-3 minutes per side, or until the tortilla is golden brown and the cheese is fully melted. Repeat this process with the remaining tortillas and filling ingredients.
06 - Allow the cooked quesadillas to rest for 1 minute before cutting them into wedges. Serve immediately with additional special sauce on the side for dipping.