Preheat oven to 350°F. Cut the top quarter of each pita and tear these into pieces and set aside. Wrap pitas in a stack in foil. Combine pita pieces, onion, lemon juice, salt, pepper, oregano, and garlic in a food processor. Process until a smooth paste forms, about 30 seconds. Transfer to a large bowl. Add beef to large bowl with paste and mix until thoroughly combined with hands or a spatula. Turn beef mixture out on to a cutting board. Form into a uniform loaf and cut into 12 sections. Roll each section into a ball and flatten into a patty about ½â thick. Place the pitas wrapped in foil in the oven and heat for 10 â 15 minutes. Heat oil in a large skillet over medium-high heat. Add patties and cook until well-browned and a dark crust forms on one side, 3 to 4 minutes. Flip the patties and cook until the second side is also well-browned. Remove to paper towels to drain. Serve with warmed pitas, gyro sauce (recipe below), and optional garnishes if desired such as shredded lettuce, sliced cucumbers, sliced tomatoes, sliced sweet onion, Kalamata olives, or feta cheese. Dill Sauce: Combine all ingredients in a small bowl. Chill 30 minutes to blend flavors.