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 the cut potato strips under cold running water to remove excess starch. Pat thoroughly dry with paper towels or a clean kitchen towel.
03 - Carefully lower the potatoes into the hot oil in small batches. Fry for 4 to 5 minutes until lightly golden and tender inside. Remove with a slotted spoon and drain on paper towels.
04 - Return the blanched fries to the hot oil for an additional 2 to 3 minutes until deeply golden and crispy. Drain again and immediately season with salt while still hot.
05 - While the fries cook, melt the butter in a large skillet over medium-low heat. Add the diced onions and cook, stirring frequently, for 12 to 15 minutes until deeply browned and fully caramelized. Remove from heat and set aside.
06 - 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 use.
07 - Arrange the crispy fries in a single layer on a baking tray. Lay cheese slices evenly across the top. Broil on high for 1 to 2 minutes until the cheese is fully melted and bubbly.
08 - Scatter the caramelized onions generously over the melted cheese. Drizzle with the prepared special sauce. Serve immediately while hot and crispy.