01 - Preheat oven to 425°F if baking. Toss the potato strips with vegetable oil and a generous pinch of salt. Spread in a single layer on a baking sheet. Bake for 25–35 minutes, flipping halfway through, until golden and crispy. Alternatively, deep-fry in hot oil at 350°F in batches for 4–6 minutes until golden, then drain on paper towels and season with salt while hot.
02 - Melt the butter in a skillet over medium heat. Add the diced onion and sauté, stirring frequently, until deeply golden brown and caramelized, about 10–15 minutes. Add a splash of water if the onions begin to stick.
03 - In a bowl, whisk together the mayonnaise, ketchup, sweet pickle relish, yellow mustard, white vinegar, granulated sugar, paprika, and black pepper until smooth and well combined. Refrigerate until ready to use.
04 - Arrange the hot fries on a serving platter. Sprinkle with shredded cheese and place under the broiler for 1–2 minutes to melt, if desired. Top with the caramelized onions and generous spoonfuls of the Animal Style sauce. Serve immediately while hot.