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 cut potato sticks in a bowl of cold water to remove excess starch. Drain thoroughly and pat completely dry with clean kitchen towels.
03 - Carefully lower potatoes into the hot oil in small batches. Fry for 4 to 5 minutes until lightly golden but still pale. Remove with a slotted spoon and drain on paper towels.
04 - Return blanched fries to the 350°F oil in batches. Fry for an additional 2 to 3 minutes until deeply golden and crisp. Transfer to paper towels and season immediately with salt.
05 - Melt butter in a skillet over medium-low heat. Add diced onions and cook, stirring frequently, until deeply browned and caramelized, about 12 to 15 minutes. Remove from heat and set aside.
06 - In a small bowl, whisk together mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, sugar, and paprika until smooth. Refrigerate until ready to use.
07 - Arrange hot fries on a baking tray in an even layer. Lay cheese slices over the top. Broil on high for 1 to 2 minutes until cheese is fully melted and bubbling.
08 - Scatter caramelized onions evenly over the melted cheese. Drizzle generously with special sauce. Serve immediately while hot and crisp.