Peel and grate 1 cucumber, then squeeze it to remove excess water. Stir in yogurt, 1/2 teaspoon lemon juice, one third of garlic, and 1/4 teaspoon each of salt and pepper to make tsatsiki.
Cut remaining cucumber into 1/4-inch pieces and stir together with tomatoes, feta (if using), 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, 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 on warm bread and top with chicken and some of lettuce and salsa. Serve remaining lettuce, salsa, and tsatsiki on the side.
Cooks' note: Tsatsiki can be made 1 day ahead and chilled.