Chicken Gyros with Cucumber Salsa and Tzatziki

2 cucumbers, divided

1 1/2 cups Greek yogurt (3/4 pound)

1 teaspoon fresh lemon juice, divided

5 garlic cloves, minced, divided

Salt and pepper

1 pint grape tomatoes, quartered

1 small red onion, halved and thinly sliced

1/3 cup chopped flat-leaf parsley

1/4 cup chopped mint

1/4 cup extra-virgin olive oil

1 rounded teaspoon dried oregano

1 rounded teaspoon dried rosemary, crumbled

1 (12-ounce) package naan bread (four 8-inch pieces) or 4 (8-inch) pocketless pita rounds

1/2 roast chicken, skin discarded, meat shredded (about 2 1/4 cups), and carcass reserved for stock

1/2 head iceberg lettuce, thinly sliced

Preheat broiler.

Peel and grate one cucumber, then squeeze it with your hands to remove excess water. Stir together with yogurt, 1/2 teaspoon lemon juice, one third of the garlic and 1/4 teaspoon each of salt and pepper to make tzatziki.

Cut remaining cucumber into 1/4-inch pieces and stir together with tomatoes, onion, parsley, mint, remaining 1/2 teaspoon lemon juice, and 1/4 teaspoon each of salt and pepper to make salsa.

Gently simmer oil, oregano, rosemary, remaining garlic, 1/4 teaspoon salt, and 1/8 teaspoon pepper in a small heavy saucepan, stirring constantly, until garlic is fragrant but not browned, one to two minutes. Toss chicken with 3 tablespoons garlic oil and brush one side of bread with remainder.

Heat bread, oiled side up, in a four-sided sheet pan, covered with foil, 3 to 4 inches from broiler three minutes. Uncover and broil, rotating bread for even coloring, until golden in spots, about two minutes.

Spread some tzatziki on warm bread and top with chicken and some of the lettuce and salsa. Serve remaining lettuce, salsa and tzatziki on the side.