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