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: Turn all burners to high, cover, and heat grill until very hot, about 15 minutes. Scrape grate clean with grill brush; dip wad of paper towels in vegetable oil and holding wad with tongs, wipe cooking grate. Leave primary burner on high and reduce other burner(s) to medium-high.

5. Cut top quarter off each pita bread. Tear one quarter into 1-inch pieces, and discard remaining three top quarters. (You should have about 1/4 cup pita pieces.) Process onion, lemon juice, salt, pepper, oregano, garlic, and pita bread pieces in food processor until smooth paste forms, scraping down sides of workbowl as necessary, about 30 seconds. Transfer onion mixture to large bowl; add lamb and gently mix with hands until thoroughly combined. Divide mixture into 4 equal pieces and roll into logs. Gently flatten logs into rectangular patties, about 1/2 inch thick, 2 1/2 inches wide, and 7 inches long.

6. Place patties on hotter side of grill. Cook, with lid down, turning once using spatula, until well browned and crust forms on each side, 8 to 10 minutes. Transfer patties to plate. While patties rest, place pita in single layer on hotter side of grill. Cook, turning once, until each pita is thoroughly warmed and faint grill marks appear, 30 to 40 seconds. Remove pita from grill and wrap tightly with aluminum foil.

7. Using soupspoon, spread 1/4 cup Tzatziki Sauce inside each pita. Put 1 patty in each pita and top with tomato slices, 1/2 cup shredded lettuce, and 2 tablespoons feta. Serve immediately.