For the sauce: Peel a clove of garlic with a crack of knife against the heel of your hand. In a small bowl, whisk up yogurt, tahini, the juice of 1 lemon and cumin. Then grate the garlic into the sauce with a zester or paste it with a knife (add a fat pinch of salt). Whisk sauce again.
For the burgers: Preheat a cast-iron skillet or heavy griddle or frying pan over medium to medium-high heat.
Place ground lamb or chicken in a mixing bowl. Season the meat with salt and pepper, cumin, coriander, paprika, sumac if using, turmeric, cardamom, crushed red pepper, allspice and nutmeg. Crack 2 cloves garlic and grate them into the bowl or finely chop. Peel the onion and grate about 3 tablespoons of it into the bowl. Finley chop the mint and parsley then add them to the lamb. Combine all the ingredients and mound up. Score meat into 4 equal portions and form 4 patties, thinner at the center for even cooking.
To prepare the burgers, add EVOO, 1 slow turn of the pan, to skillet (a thin layer to just glaze the surface). Add patties and cook about 7 minutes for lamb, a minute or two longer for chicken, turning occasionally. If cooking lamb, press down with spatula when you turn the patties -- lean ground lamb tends to tighten as it cooks more so than other ground meats.
When you remove the patties they can be served as such but for a more authentic facsimile of shawarma, we rest the patties and very thinly slice them on a bias. Regardless, while the burgers cook, prepare your toppings of choice.
Assemble burgers: bun bottoms, sauce, patty or thinly sliced shawarma, lettuce, hot cherry peppers, cucumbers, herb leaves, dressed fennel and onions, bun top. OR split a pita and fill with giardiniera and Israeli pickles.
Poke with a couple of toothpicks and split burgers down the middle. Serve burgers with chips.