Whisk together mayonnaise, ketchup, pickle relish, diced red onion, white vinegar, and garlic powder. Cover and refrigerate; this can be made up to one week ahead.
Press each 1.2 oz portion of ground beef onto the tortillas, spreading thin to the edges. Spread a small amount of yellow mustard on top and season with salt and pepper.
Heat a cast iron pan until smoking hot. Place tortillas patty-side down and use a spatula to press the meat flat, creating a crispy crust.
After about a minute, scrape the patty and flip the tortilla briefly to warm the other side. Remove and prepare for cheese.
Immediately top each cooked patty with a slice of cheese so it melts perfectly.
Return the pan to heat and sauté diced onions in the rendered beef fat until soft and lightly browned, about 8 minutes.
Spread a spoonful of sauce on each patty, then layer grilled onions, pickles, tomato, and shredded lettuce. Serve hot.