1. For the Tzatziki Sauce: Line fine-mesh strainer set over deep container or bowl with 3 paper coffee filters or triple layer of paper towels. Spoon yogurt into lined strainer, cover, and refrigerate for 30 minutes.

2. Meanwhile, combine cucumber, 1/8 teaspoon salt, and lemon juice in colander set over bowl and let stand 30 minutes.

3. Discard drained liquid from yogurt. Combine thickened yogurt, drained cucumber, remaining 1/4 teaspoon salt, garlic, and mint in clean bowl.

4. For the Patties: Adjust oven rack to middle position and heat oven to 350°F. Cut top quarter off each pita bread. Tear quarters into 1-inch pieces. (You should have 3/4 cup pita pieces.) Stack pitas and tightly wrap with aluminum foil. Process onion, lemon juice, salt, pepper, oregano, garlic, and pita bread pieces in food processor until smooth paste forms, about 30 seconds. Transfer onion mixture to large bowl; add beef and gently mix with hands until thoroughly combined. Divide mixture into 12 equal pieces and roll into balls. Gently flatten balls into round disks, about 1/2-inch-thick and 2 1/2 inches in diameter.

5. Place foil-wrapped pitas directly on oven rack and heat for 10 minutes. Meanwhile, heat oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add patties and cook until well browned and crust forms, 3 to 4 minutes. Flip patties, reduce heat to medium, and cook until well browned and crust forms on second side, about 5 minutes longer. (See below for tips on flipping patties.) Transfer patties to paper towel-lined plate.

6. Using soupspoon, spread 1/4 cup Tzatziki Sauce inside each pita. Divide patties evenly among pitas; top each sandwich with tomato slices, 1/2 cup shredded lettuce, and 2 tablespoons feta. Serve immediately.

Serves 4