Preheat the oven to 220°C (425°F) and line a baking sheet with parchment paper.
Pat the tofu dry. Shave into thin slices, 2 to 4 mm thick, using a vegetable peeler, sharp knife, or mandoline.
Toss the tofu in a bowl with olive oil, yogurt, cornstarch, soy sauce, and all the spices until coated.
Spread in a single layer on the baking sheet. Bake for 20 to 25 minutes, flipping halfway, until crispy and golden.
While the tofu bakes, stir together the yogurt sauce ingredients. Toss the salad ingredients with salt and pepper.
Warm the flatbreads in a dry skillet for 30 seconds per side. Spread with sauce, top with tofu and salad, drizzle with more sauce, and wrap.
Notes
Cornstarch and soy sauce are optional. Swap the spice blend for store-bought shawarma seasoning if preferred. Air fry at 200°C (400°F) for 12 to 15 minutes as an alternative to baking.