01 - Peel the russet potatoes and cut them into uniform 1/4-inch thick sticks. For maximum crispiness, soak the cut potatoes in cold water for 30 minutes, then drain and pat completely dry with a clean kitchen towel.
02 - For baking: Preheat oven to 425°F. Toss the dried potato strips with vegetable oil and a generous pinch of kosher salt. Spread in a single layer on a baking sheet and bake for 25 to 35 minutes, flipping halfway through, until golden brown and crispy. For frying: Heat 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 season with salt while still hot.
03 - Melt the unsalted butter in a large skillet over medium heat. Add the finely diced onion and sauté, stirring frequently, until deeply golden brown and fully caramelized, about 10 to 15 minutes. Add a splash of water if the onions begin to stick to the pan.
04 - In a small mixing bowl, whisk together the mayonnaise, ketchup, sweet pickle relish, yellow mustard, distilled white vinegar, granulated sugar, paprika, and black pepper until smooth and well combined. Refrigerate until ready to use.
05 - Arrange the hot crispy fries on a large serving platter. Sprinkle the shredded cheese evenly over the top and place under a broiler for 1 to 2 minutes until melted and bubbly. Remove from the broiler, top generously with the caramelized onions, and drizzle with spoonfuls of the Animal Style sauce. Serve immediately while hot.