Preheat the oven to 325 degrees F.
To make the tzatziki sauce, grate the cucumber on the large holes of a box grater onto a clean dishtowel and then use the towel to squeeze out the extra liquid. Put the cucumber a medium bowl and finely grate 1 clove of the garlic over it (use only half if it is a large clove). Add the yogurt, 1 tablespoon of the olive oil, the dill, 1 tablespoon of the lemon juice, 1/2 teaspoon salt and some pepper. Adjust the seasonings with additional lemon juice and salt. Chill the tzatziki while making and cooking the meat to allow the flavors to blend (you can also make this up to a day ahead of time).
Add the ground meat, rosemary, oregano and lemon zest to the bowl of a food processor. Chop the remaining 2 cloves of garlic and add to the bowl with 1 teaspoon salt, 1/4 teaspoon pepper and the pepper flakes, if using. Process this mixture until it is well combined and fine in texture but not pasty, stopping to scrape down the bowl as needed.
Form the meat into eighteen 2- by 4-inch, very thin oval-shape patties (they don't need to be perfect looking). Heat 1 tablespoon of the oil in a large skillet set over medium-high heat. Add 6 of the patties and cook until well browned on each side and just cooked through, about 3 minutes. Transfer the patties to a rimmed baking sheet and place in the oven to keep warm. Repeat, cooking the remaining patties in two batches, using the remaining 2 tablespoons of oil as needed.
Put the pitas in the oven to warm, about 3 minutes. Then top each pita with 3 meat patties, shredded lettuce, sliced tomato and red onions. Top with about 1/4 cup tzatziki sauce. Wrap and enjoy.