Whisk together all the chicken marinade ingredients in a bowl. Thinly slice chicken going against the grain and place in the marinade. Mix and set aside.
Boil spaghetti noodles in salted water until al dente. Drain and rinse under cold water for a minute.
While the water boils for the spaghetti, prepare your vegetables. Be sure to string the sugar snap peas on both sides. Thinly slice the cabbage, you'll only use about 1/4 of a medium sized head. Mince the garlic, slice the green onions, yellow onion and shitake mushrooms. Cut the broccoli into bite size pieces, you'll use about 2 crowns of broccoli.
Heat 2 tablespoons of vegetable oil in a wok over high heat for 2 minutes. Add half the chicken. Cook on both sides until all pink is gone and the outside is golden brown, about 5 minutes. Place cooked chicken on a plate and cook the second batch of chicken in the wok. Remove chicken from the wok and wipe down if needed.
Heat 1/4 c vegetable oil on high in the same wok for 2 minutes. Add the yellow onions, broccoli, cabbage, matchstick carrots and sugar snap peas. Stir fry for 6 minutes, before adding the minced garlic, sliced green onions and sliced shitake mushrooms. Season with 1 tsp kosher salt .Stir fry another 6 minutes, stirring almost constantly.
Turn heat down to medium. Add the spaghetti and sauce to the wok, toss with tongs.
Add the chicken to the wok and toss with tongs until everything is coated and the chicken is heated through. Serve hot.