01 - Preheat vegetable oil in a deep fryer or heavy pot to 350°F.
02 - Rinse cut potato fries in cold water and pat thoroughly dry. Fry in batches for 4–5 minutes until lightly golden. Remove and drain on paper towels.
03 - Return par-cooked fries to the hot oil and fry an additional 2–3 minutes until deeply golden and crisp. Season immediately with salt.
04 - Melt butter in a skillet over medium-low heat. Add diced onions and cook, stirring frequently, until deeply browned and caramelized, about 12–15 minutes. Set aside.
05 - Whisk together mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, sugar, and paprika until fully combined. Refrigerate until ready to use.
06 - Arrange fries on a baking tray. Layer cheese slices over the top and broil for 1–2 minutes until the cheese is fully melted and bubbling.
07 - Scatter caramelized onions evenly over the melted cheese. Drizzle generously with special sauce and serve immediately.