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. Working in small batches to avoid overcrowding, carefully lower the potatoes into the hot oil and fry for 4 to 5 minutes until lightly golden and just cooked through. Remove with a slotted spoon and drain on a wire rack set over paper towels.
03 - Return the blanched potatoes to the 350°F oil in batches and fry for an additional 2 to 3 minutes until deeply golden and crisp. Transfer back to the wire rack and immediately season generously with kosher salt while still hot.
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, softened, and fully caramelized, 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 serve.
06 - Arrange the double-fried potatoes in an even layer on a baking tray. Lay the cheese slices over the hot fries. 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 and crisp.