Whisk together the yogurt, vinegar, 1/4 cup of olive oil, lemon juice, garlic, cinnamon, nutmeg, oregano, and bay leaf in a bowl, and pour into a resealable plastic bag. Add the thinly sliced lamb, coat with the marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator 6 hours to overnight.
Heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Pour in the meat and marinade, and cook until the lamb is no longer pink, and is tender, 15 to 20 minutes. Stir frequently as the meat cooks. Divide the cooked lamb meat among the warmed pita breads, and garnish with tomatoes, onion, and fresh mint leaves to serve.