01 - Pour vegetable oil into a deep fryer or heavy-bottomed 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 running water to remove excess starch. Pat thoroughly dry with clean kitchen towels or paper towels.
03 - Carefully lower the dried fries into the hot oil in small batches. Fry for 4 to 5 minutes until lightly golden and just cooked through. Remove with a slotted spoon and drain on paper towels.
04 - Return the blanched fries to the hot oil for an additional 2 to 3 minutes until deeply golden and crisp. Transfer to a paper towel–lined tray and season immediately with salt.
05 - 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 fully caramelized. Set aside.
06 - In a small mixing bowl, whisk together the mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, sugar, and paprika until smooth and well combined. Refrigerate until ready to use.
07 - Arrange the crispy fries in an even layer on a baking tray. Lay cheese slices over the top and broil for 1 to 2 minutes until the cheese is fully melted and bubbling. Scatter the caramelized onions over the cheese and drizzle generously with the special sauce. Serve immediately.