01 - Preheat vegetable oil in a deep fryer or heavy saucepan to 350°F.
02 - Rinse cut potatoes in cold water and pat thoroughly dry. Fry in batches for 4-5 minutes until lightly golden. Remove with a slotted spoon and drain on paper towels.
03 - Return par-cooked fries to the hot oil for an additional 2-3 minutes until deeply golden and crisp. Drain again and season immediately with salt.
04 - While the fries cook, melt butter in a skillet over medium-low heat. Add diced onions and cook, stirring frequently, until deeply browned and caramelized, about 12-15 minutes. Set aside.
05 - In a small mixing bowl, combine mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, sugar, and paprika. Stir until smooth and refrigerate until ready to use.
06 - Arrange the crispy fries on a baking tray. Layer cheese slices evenly over the top and broil for 1-2 minutes until fully melted and bubbling.
07 - Scatter caramelized onions generously over the melted cheese. Drizzle with special sauce and serve immediately while hot.