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, until golden and crispy. Alternatively, deep-fry in hot oil at 350°F in batches until golden, 4–6 minutes, and drain on paper towels. 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, sugar, paprika, and black pepper until smooth. 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 to melt, if desired. Top with the caramelized onions and generous spoonfuls of the Animal Style sauce. Serve immediately while hot.