Whisk rice vinegar, 1 tablespoon vegetable oil, sesame oil, brown sugar, soy sauce, and ginger together in a large bowl until dressing is combined.

Bring water to a boil in a large pot. Add soba noodles, stir, and return water to a boil. Boil noodles until tender, 4 to 5 minutes. Drain noodles in a colander under cold running water until cool, about 1 minute.

Heat 2 teaspoons vegetable oil in a skillet over medium heat. Cook chicken breast pieces until no longer pink in the center and the juices run clear, 2 to 4 minutes. Add garlic, salt, and pepper; stir until fragrant, about 1 minute more.

Toss soba noodles, chicken, celery, carrot, red pepper, cilantro, green onion, and sesame seeds together with dressing in large bowl.