In a small bowl, whisk together mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, paprika, garlic powder, and onion powder. Taste and adjust as needed.

Heat a skillet over medium-high heat. Add the ground beef and chopped onion, breaking the meat into small pieces with a spatula. Season with salt and pepper. Cook until the beef is deeply browned and the onions are soft, about 7 to 9 minutes.

Warm a clean skillet over medium heat and add a pat of butter. Place a tortilla in the skillet for 20 seconds on each side just to soften and add a slight golden color. Repeat with remaining tortillas.

Spread 1 to 2 tablespoons of the sauce on each warm tortilla. Spoon a quarter of the beef mixture down the center of each wrap. Top with a slice of cheese, a sprinkle of shredded lettuce, and a scattering of diced pickles.

Fold the bottom of the tortilla up and the sides in, rolling into a snug wrap. If you like the cheese extra melty, place assembled wraps seam-side down on the skillet for 1 to 2 minutes until the cheese loosens and the outside is crisp.

Slice each wrap in half on the diagonal and serve immediately with extra sauce for dipping.