01 - Preheat oven to 425°F. 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 oil heated to 350°F in batches for 4–6 minutes until golden, then drain on paper towels and 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 to the pan.
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. Refrigerate until ready to use.
04 - Arrange the hot fries on a serving platter. Sprinkle with shredded cheese and place under a broiler for 1–2 minutes until melted, if desired. Top with the caramelized onions and generous spoonfuls of the Animal Style sauce. Serve immediately while hot.