Whisk soy sauce, corn starch, and sesame oil together in a large bowl until smooth; add chicken strips and toss to coat. Cover and refrigerate for at least 20 minutes.

Combine chicken broth, oyster sauce, and sugar in a small bowl and set aside.

Bring a large pot of water to a boil. Add noodles and cook over medium heat until cooked through but still firm to the bite, 4 to 5 minutes. Drain and rinse with cold water.

Heat vegetable oil in a large skillet. Cook and garlic in hot oil for 30 seconds; add marinated chicken. Cook and stir until browned and no longer pink in the center, 5 to 6 minutes. Remove chicken mixture to a plate. Cook and stir bok choy, zucchini, snap peas, and carrot in the hot skillet until softened, about 2 minutes. Return noodles and chicken mixture to the skillet. Pour broth mixture into noodle mixture; cook and stir until warmed through, about 2 minutes. Serve garnished with green onions.