Bring about 3 quarts of water to a boil in a large saucepan.
Julienne the cucumber and place it in a large bowl along with the vinegar, sesame oil and soy sauce. Let the cucumber marinate while you prepare the rest of the vegetables.
Add about ½ teaspoon salt to the water then stir in the soba noodles. Set the timer for five minutes. Continue stirring the sobas until the water comes back to the boil. Lower the heat so the water is at a medium boil. After one minute, stir in the green beans. When the timer goes off, test a noodle to be sure it is cooked. Pour the sobas and beans into a colander and rinse with a little cold water to stop the cooking process. Drain the noodles and beans very well as water will dilute the dressing.
Stir the well-drained noodles and beans into the bowl with the cucumbers. When coated with dressing, stir in the mint, scallions and carrots. Serve sprinkled with the toasted sesame seeds.