For Tzatziki Sauce: Spoon yogurt into lined strainer, cover, drain in refrigerate for 30 minutes.

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

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

For Patties: Heat grill until very hot. Wipe cooking grate with oil.

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 rectangular patties, about 1/2 inch thick.

Grill patties with lid down, turning once, until well browned and crust forms on each side, 8 to 10 minutes. While patties rest, place pita in single layer on hotter side of grill. Cook, turning once, until each pita is warmed and grill marks appear, 30 to 40 seconds. Remove pita and wrap tightly with foil.

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.