01 - Pour vegetable oil into a deep fryer or heavy pot to a depth of at least 3 inches. Heat to 350°F over medium-high heat.
02 - Rinse the cut potato fries under cold water to remove excess starch, then pat thoroughly dry with paper towels. Fry in batches for 4 to 5 minutes until lightly golden but not yet crisp. Transfer to a paper towel–lined plate and let drain.
03 - Return the par-cooked fries to the hot oil in batches and fry for an additional 2 to 3 minutes until deeply golden and crisp. Remove with a slotted spoon, drain on fresh paper towels, and immediately season with salt.
04 - While the fries cook, melt the butter in a large skillet over medium-low heat. Add the diced onions and cook, stirring frequently, for 12 to 15 minutes until deeply browned and caramelized. Remove from heat and set aside.
05 - In a small mixing bowl, combine the mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, sugar, and paprika. Stir until smooth and well blended. Refrigerate until ready to serve.
06 - Arrange the finished fries in a single layer on a baking tray. Lay cheese slices evenly over the top and broil on high for 1 to 2 minutes, watching closely, until the cheese is fully melted and bubbling.
07 - Scatter the caramelized onions generously over the melted cheese. Drizzle the special sauce across the top. Serve immediately while hot and crisp.