For the chicken: Put the chicken pieces in a large bowl. Add the olive oil, paprika, 1/2 teaspoon salt, yogurt, crushed garlic and soy sauce and toss to coat well. Add the lemon juice and toss again. Cover with plastic wrap and refrigerate for at least 1 hour or up to overnight.
For the flatbread: Combine the flour and a pinch salt in a stand mixer fitted with the dough hook. With the machine on low, add 1/2 cup water and mix until just combined. Increase the speed to medium-low and continue kneading, adding more water as necessary, just until the dough comes together into a ball. Increase the speed to medium and knead for another 30 seconds.
Flour a work surface and transfer the dough to it (the dough will be sticky). Sprinkle more flour over the dough and knead until smooth. Portion the dough into 6 pieces, roll into balls and flatten with a rolling pin. Brush off the excess flour with a clean, dry pastry brush.
Heat a large nonstick skillet or griddle over medium-high heat. Add one piece of dough and cook until the first side is lightly bubbled and golden, a few minutes, then flip and cook the other side. Remove from the heat and wrap in a clean kitchen towel to soften and keep warm. Repeat with the remaining dough.
For the toumeyyaeh (garlic sauce): Mix 1 cup water with the cornstarch in a microwave-safe bowl until dissolved. Microwave for 2 minutes until the mixture is very thick like pudding.
Put the garlic, cornstarch mixture, mayonnaise, olive oil and a few pinches of salt in a blender or mini chopper and process until smooth. Let cool to room temperature, cover and refrigerate.
Preheat a grill pan over medium-high heat. Brush the pan with olive oil and grill the chicken until marked both sides and cooked through, 7 to 10 minutes. Remove from the grill and let rest for 5 minutes.
Chop the chicken and add to a bowl with the pickles, onions, 3 tablespoons of the garlic sauce and sumac. Toss to combine.
Place each flatbread on a piece of parchment or waxed paper and spread with some of the extra garlic sauce. Fill each with some of the chicken mixture and a few french fries and parsley, if using. Roll the flatbread in the paper, slice in half on a bias and serve.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)