01 - Pour vegetable oil into a deep fryer or heavy-bottomed pot to a depth of at least 3 inches. Heat to 350°F over medium-high heat.
02 - Rinse cut potato strips 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 just beginning to crisp. Remove with a slotted spoon and drain on fresh paper towels.
03 - Return the blanched fries to the hot oil in batches and fry for an additional 2 to 3 minutes until deeply golden and fully crisp. Transfer to paper towels and season immediately with 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 and richly caramelized, approximately 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 fully incorporated. Refrigerate until ready to use.
06 - Arrange the crispy fries in a single layer on a baking tray. Layer the cheese slices evenly over the top and broil on high for 1 to 2 minutes until the cheese is fully melted and bubbling.
07 - Scatter the caramelized onions generously over the melted cheese. Drizzle the special sauce across the top. Serve immediately while hot and crispy.