In a heavy, 5-quart Dutch oven or pot fitted with a deep-fry or candy thermometer, heat oil over medium-high heat until thermometer registers 375°.
Meanwhile, in a small bowl, whisk mayonnaise, ketchup, relish, vinegar, mustard, and ¼ teaspoon salt to combine.
In a large skillet over medium-high heat, melt butter. Add onion and ¼ teaspoon salt and cook, stirring occasionally, until onion is well browned and translucent, 8 to 10 minutes. Transfer onions to a small bowl; set aside.
Using a metal spider, lower half of fries into oil and cook, stirring to keep fries from sticking together, until golden brown, 4 to 5 minutes. Using spider, transfer fries to a paper towel-lined sheet tray; season with a pinch of salt. Repeat with remaining fries.
In a small heatproof bowl, microwave cheese and milk in 30-second increments, whisking in between, until cheese is melted and smooth, about 3 minutes.
Top fries with cheese mixture, then cooked onions, and finish with animal sauce.
