In a covered medium saucepan, cook green beans in boiling water about 4 minutes or until crisp-tender. Drain. Rinse under cold water; drain again.
Place green beans in a large bowl. Stir in tuna, the 1 cup cherry tomatoes, the 1/3 cup olives, the onion, and the 2 tablespoons mint. Add lemon juice, oil, and pepper; toss to combine. Stir in mesclun.
To serve, cut pita flatbreads in half crosswise. Cut each pita half in half horizontally. Fill each pita half with about 1/2 cup of the tuna mixture. On each of the wooden skewers, thread a whole cherry tomato, a whole olive, and a mint leaf. Spear each serving with a skewer.