Make the meat mixture. In a large bowl, mix the ground lamb, sirloin, lemon zest, garlic, minced red onion, parsley, spices, salt and pepper together. Use your hands to mix everything together until it’s well combined and then cover and refrigerate for at least 2 hours, up to 48.
Make the white sauce. In a small bowl, whisk together the yogurt, mayonnaise, lemon zest, garlic, and cumin and season with salt and pepper. Whisk in just enough water to make it drizzle-able and then refrigerate until you’re ready to serve. You can store this in the fridge for up to 5 days.
Marinate the onions. An hour before you plan to cook, mix the thinly sliced red onion with the lemon juice, parsley, olive oil, and sumac and season with salt and pepper. Toss the onion well to coat and let them sit at room temperature for 1 hour. Store in an airtight container in the fridge for up to 3 days.
Press the meat into the pita. Divide the meat mixture into 4 oz portions and roll each portion into a ball. Press the meat in an even layer on the flatter side of the pita so that it extends to the edges. Sprinkle the meat with more za’atar.
Cook the smash burgers. Preheat a large griddle or skillet over medium heat for at least 5 minutes. Drizzle a small amount of oil into the pan and press the pita into the skillet, meat side down. Cook for 4-5 minutes, pressing down occasionally, until the lamb is browned and crispy. Remove the lamb pitas from the griddle, wipe out any excess oil, and return them to the pan lamb side up to toast the pita for 1 minute.
Assemble. Transfer the lamb pitas to a serving platter and top as desired with the sumac onions, garlic yogurt sauce, lettuce, tomatoes, and feta.