01 - Preheat the oven to 425°F if baking. Toss the potato strips with vegetable oil and a generous pinch of salt. Spread them in a single layer on a baking sheet, making sure they do not overlap. Bake for 25 to 35 minutes, flipping halfway through, until golden brown and crispy. Alternatively, heat frying oil to 350°F in a deep fryer or heavy pot and fry the potato strips in batches for 4 to 6 minutes until golden. Drain on paper towels and salt while still hot.
02 - Melt the butter in a large skillet over medium heat. Add the diced onion and sauté, stirring frequently, until deeply golden brown and fully caramelized, about 10 to 15 minutes. If the onions begin to stick, add a small splash of water and continue stirring.
03 - In a mixing bowl, whisk together the mayonnaise, ketchup, sweet pickle relish, yellow mustard, white vinegar, sugar, paprika, and black pepper until smooth and well combined. Cover and refrigerate until ready to use.
04 - Arrange the hot fries on a serving platter. Sprinkle the shredded cheese evenly over the top. Place under a broiler for 1 to 2 minutes to melt the cheese, if desired. Top generously with the caramelized onions and drizzle spoonfuls of the Animal Style sauce over everything. Serve immediately while hot.