Prepare the Animal Style spread by combining mayonnaise, ketchup, sweet pickle relish, white vinegar, and sugar in a mixing bowl. Whisk together until smooth and well-blended, then refrigerate while you prepare the other components to allow the flavors to meld together.
Cut the peeled russet potatoes into 1/4-inch thick strips, keeping them as uniform as possible for even cooking. Place the cut fries in a large bowl of cold water and let them soak for at least 10 minutes to remove excess starch, which will help them become crispier when fried.
Heat the vegetable oil in a deep fryer or large heavy-bottomed pot to 325°F for the first fry. Drain the potatoes thoroughly and pat them completely dry with paper towels, as any moisture will cause dangerous splattering. Fry the potatoes in batches for 3-4 minutes until they are cooked through but still pale, then remove and drain on paper towels.
While the fries are resting, heat butter in a large skillet over medium heat until melted and foaming. Add the thinly sliced onions and cook, stirring frequently, for 12-15 minutes until they are deeply caramelized and golden brown with a sweet, savory aroma.
Increase the oil temperature to 375°F for the second fry. Working in batches again, fry the par-cooked potatoes for 2-3 minutes until they turn golden brown and crispy, listening for the vigorous bubbling sound that indicates proper crisping. Remove with a slotted spoon and immediately season with salt while still hot.
Arrange the hot, crispy fries on a large serving platter or individual plates in an even layer. Tear the American cheese slices into smaller pieces and distribute them evenly over the hot fries, allowing the residual heat to begin melting the cheese.
Generously drizzle the Animal Style spread over the cheese-covered fries in a zigzag pattern, using about 3-4 tablespoons or to taste. The spread should create creamy pockets throughout the fries.
Top the fries with the caramelized grilled onions, distributing them evenly across the entire dish. Serve immediately while the fries are still hot and crispy, and the cheese is melted and gooey, capturing the authentic In-N-Out experience.
