Combine 3 tablespoons soy sauce and chicken in a zip-top plastic bag; seal and marinate in refrigerator 30 minutes, turning occasionally.
Heat 1 teaspoon oil in a large nonstick skillet or wok over medium-high heat. Add chicken; cook 3 minutes or until done. Remove chicken from skillet; keep warm. Add 1 teaspoon oil, onion, and carrot to skillet; stir-fry 2 minutes. Add cabbage; stir-fry 2 minutes or until cabbage begins to wilt.
Combine 1/4 cup soy sauce, sake, ginger, sugar, and salt. Pour sake mixture over cabbage mixture. Add chicken and soba; toss well to coat. Cook 1 minute or until thoroughly heated.