Gather the ingredients.
To make the meat mixture, thoroughly combine the ground lamb, red onion, garlic, lemon juice, salt, cumin, pepper, oregano, and nutmeg in a mixing bowl.
Divide into four equal portions and shape into oblong patties about 3-inch wide by 6-inch long by 1/2-inch thick. Notice that the patty is shaped more like a sausage (long and thin) rather than a hamburger patty. This is intentional, to ensure the patty fits in the flatbread.
Refrigerate for one hour.
Preheat a grill or grill pan over high heat. Alternatively, heat a large nonstick skillet over high heat.
If grilling, brush the grill or grill pan with oil. If pan-frying, add a dollop of oil to the pan. Grill or pan-fry the patties for 3 to 4 minutes per side, or until cooked through with crispy edges.
To assemble the sandwiches, spread tzatziki sauce down the center of a flatbread round or in the pocket of the pita.
Add a lettuce leaf, some sliced tomato, and a few thin onion slices.
Add the patty and fold the bread over the lamb. Repeat with the remaining sandwiches.