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 potatoes in cold water to remove excess starch, then pat thoroughly dry with paper towels. Fry in small batches for 4 to 5 minutes until lightly golden but still pale. Transfer to a paper towel–lined tray and let cool.
03 - Return the blanched fries to the 350°F oil and fry for an additional 2 to 3 minutes until deeply golden and crisp. Drain on fresh paper towels and season immediately 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, until deeply browned and caramelized, about 12 to 15 minutes. 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 use.
06 - Arrange the hot crispy fries in an even layer on a baking tray. Lay cheese slices 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 evenly over the melted cheese. Drizzle generously with the special sauce. Serve immediately while hot.