In a large bowl, whisk together the olive oil, lemon juice, vinegar, tomato paste, garlic, cumin, paprika, coriander, onion powder, salt, pepper, turmeric, cinnamon, allspice, cayenne, and parsley.
Add the sliced beef and toss until thoroughly coated. Cover and refrigerate for at least 4 hours or overnight.
To prepare the garlic sauce, whisk together the mayonnaise, Greek yogurt, garlic, lemon juice, and salt until smooth. Refrigerate until ready to use.
Heat a large cast-iron skillet over high heat. Add the neutral oil.
Cook the beef in batches, spreading it into a single layer. Cook for 2 to 3 minutes per side until deeply browned with lightly charred edges.
Transfer the beef to a plate and let it rest for 2 to 3 minutes.
Warm the pita breads or flatbreads.
Spread a generous amount of garlic sauce down the center of each wrap.
Layer with lettuce, tomatoes, cucumber, red onion, pickled turnips, dill pickles, parsley, and the cooked beef.
Fold in the sides and roll tightly.
For a shawarma-shop style finish, toast the wraps seam-side down in a dry skillet for about 1 minute per side.
Serve immediately.