Soak 8 bamboo skewers (8 in.) in water for half an hour.
Heat a grill to medium (350° to 450°). Coarsely grind pistachios in a food processor, then pour into a bowl. Whirl chopped onion in processor until smooth and add to nuts along with egg white, meat, baharat, black pepper, 2 1/2 tsp. Aleppo pepper, and the salt. Mix with your hands. Shape into 4 patties, 1/2 in. thick.
Thread tomatoes lengthwise onto skewers. Combine mayonnaise, 1 to 2 tsp. Aleppo pepper, and the lemon juice and set aside. Brush onion slices with oil.
Grill burgers, tomatoes, and onion slices, turning once, until burgers are cooked through (6 to 8 minutes total), and vegetables are charred (3 to 5 minutes). Transfer to a platter as done. Grill buns cut side down until golden, 1 to 2 minutes.
Spread buns with mayo. Toss onions with a little Aleppo pepper. Build burgers with lettuce, onions, and some tomatoes from skewers; serve the rest on the side.
*Find in the spice aisle or at worldspice.com