Bring a large pot of lightly salted water to a boil. Cook linguine in the boiling water until cooked through yet firm to the bite, 6 to 9 minutes; drain. Toss with sesame oil in a bowl to coat.
Mix water, soy sauce, cornstarch, and freshly grated ginger into a small bowl.
Heat 1 tablespoon vegetable oil in a large skillet or wok over medium-high heat. Cook and stir chicken in hot oil until cooked through, 5 to 7 minutes. Remove chicken from skillet to a bowl and return pan to heat.
Heat remaining 1 tablespoon vegetable oil in the pan. Cook and stir bell pepper, celery, broccoli, snap peas, onion, cabbage, and garlic in oil until slightly tender, about 5 minutes. Pour prepared sauce over vegetable mixture; stirring continually, bring to a simmer and cook until slightly thickened, 30 to 45 seconds. Return chicken to the pan; cook 1 minute more. Add drained pasta; toss to coat.