Preheat broiler.

To make Tsatski sauce:

Peel and grate 2 cucumbers, then squeeze it with your hands and/or drain with a paper towel to remove excess water. Stir together with yogurt, 1/2 teaspoon lemon juice, one third of garlic, and 1/4 teaspoon each of salt and pepper.

To make cucumber-tomato salsa

Chop remaining 3 cucumbers 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.

Make garlic oil by gently simmering 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, 1 to 2 minutes. Toss chicken with 3 tablespoons garlic oil and brush one side of bread with remainder.

Heat bread, oiled side up, in a 4-sided sheet pan, covered with foil, 3 to 4 inches from broiler 3 minutes. Uncover and broil, rotating bread for even coloring, until golden in spots, about 2 minutes.

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