Warm olive oil in a skillet. Add the diced onion for the beef filling and cook until softened, then stir in the minced garlic.
Add the ground beef and break it into small crumbles as it cooks. Cook the beef through, then stir in the paprika, chili powder, garlic powder, onion powder, salt, black pepper, and tomato paste.
Pour in the 1/2 cup of water and 2-3 tbsp worcestershire. Stir and simmer until the liquid reduces to a moist coating around the beef rather than a pool in the skillet.
For the grilled onions, warm olive oil and the 2 tbsp butter in a separate skillet. Add the second diced onion and salt, then cook, stirring, until softened and browned.
For the burger sauce, stir together the mayo, ketchup, yellow mustard, relish, 1/2 tbsp worcestershire, salt, pepper, and paprika until combined.
Lay out a flour tortilla. Add cooked beef, then burger sauce, grilled onions, and shredded cheese, leaving enough tortilla around the filling to fold it closed.
Fold the sides inward, then roll the tortilla into a closed parcel with the seam underneath. Repeat with the remaining filling and tortillas.
Place the folded wraps in the skillet with the seams underneath. Cook, turning as needed, until the tortillas are browned and the cheese inside has melted.
Finish with garlic butter and parsley. Serve the wraps with the remaining burger sauce for dipping.