01 - Pour vegetable oil into a deep fryer or heavy-bottomed pot to a depth of 3 inches. Heat to 350°F over medium-high heat.
02 - Rinse cut potato fries under cold water to remove excess starch, then pat thoroughly dry with paper towels. Fry in batches for 4 to 5 minutes until lightly golden and tender inside. Drain on a wire rack set over a sheet pan.
03 - Return par-cooked fries to the 350°F oil and fry an additional 2 to 3 minutes until deeply golden and crispy. Immediately season generously with salt while still hot.
04 - While the fries cook, melt butter in a large skillet over medium-low heat. Add diced onions and cook, stirring frequently, for 12 to 15 minutes until deeply browned, softened, and caramelized. Remove from heat and set aside.
05 - In a small mixing bowl, combine mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, sugar, and paprika. Stir until smooth and fully incorporated. Refrigerate until ready to serve.
06 - Arrange 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 bubbly.
07 - Scatter caramelized onions evenly over the melted cheese. Drizzle generously with special sauce. Serve immediately while hot and crisp.