1. In a small bowl, whisk together the mayo, ketchup, relish, vinegar, mustard, and a pinch of salt. Set aside.
2. In a medium bowl, mix the ground beef with 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper. Divide the mixture into four equal portions and roll into balls. Set aside.
3. In a large skillet over medium-high heat, melt the butter. Add the onions, season with salt and pepper, and cook, stirring often, until tender and golden around the edges, 5–7 mins. Transfer to a bowl.
4. Add the peanut oil and heat to 350F, or until a potato slice sizzles. Add potatoes in an even layer and cook, stirring occasionally, until deep golden and crispy, 4–6 mins. Remove the skillet from the heat, transfer the potatoes to a paper towel-lined plate and sprinkle with salt.
5. Carefully pour off the oil, leaving a thin coating behind. Return the pan to the stove over medium-high heat. Working one at a time, add a patty and press it down firmly with a spatula to flatten (you may need to work in batches). Cook 1–2 mins until browned on the bottom, flip, top with cheese, and cook another 1–2 mins until melted and cooked through.
6. Assemble the bowls with a base of lettuce, tomato, and grilled onions. Top with burger patties, French fry crisps, and a drizzle of dressing. Enjoy!
