Do Ahead: Dry chicken breasts and cut into bite size pieces. Mix chicken with all of Chicken Mixture ingredients, making sure all chicken pieces are in contact with marinade. Marinade at least 2 hours or overnight.
Start boiling a large pot of salted water for the pasta. Heat 2 TB oil in large heavy skillet or wok. Cook chicken with marinade until chicken is no longer pink in center. Do not overcook. Transfer cooked chicken with juices to a large bowl and set aside.
In same skillet/wok, heat 2 TB oil and cook half of the chopped garlic until fragrant, about 30 seconds. Add broccoli florets and julienned carrots. Cook on medium high until just tender, adding dashes of water as needed to keep veggies from drying out. Add 2 tsp kosher salt, 1 TB oyster sauce, and 1/2 tsp black pepper. Stir fry another minute. Add sliced mushrooms and cook just until soft. Transfer veggies to the bowl with cooked chicken.
Cook spaghetti noodles just until al dente. Drain in colander; don't rinse. Heat 3 TB oil in large pot or wok with rest of chopped garlic, cooking just until fragrant. Add back the cooked noodles. Add 2 TB Asian sesame oil and 4 TB oyster sauce. Stir fry the noodles on medium about 3 minutes (easier with chopsticks.) Add salt/pepper to taste, if needed.
Stir in desired amount of cooked veggies/chicken mixture with juices (use all of it, or save some to go with rice the next day.) Stir fry thoroughly a few more minutes, until well incorporated.
Garnish with chopped scallions or cilantro if desired. Serve warm.