One note before you jump in, a lot of these individual steps can be completed at the same time (or in some cases, ahead of time), rather than linearly.

Preheat the oven to 400 F.  Cut the spaghetti squash in half (if possible) and smear the exposed part with a bit of olive oil and salt.  Roast the two halves face-down until fork-tender and you can easily shred the insides.  Or cook it any other way you prefer.  Set aside to cool (this can be done ahead of time).

Pat the chicken breasts dry and dice into bite-size pieces.  Saute in a pan on medium with a bit of olive oil and salt.  The goal is to keep them from getting soggy and watery if possible.  Add in a big spoonful of minced garlic toward the very end (don't add it too early, or it will burn).  Set aside.

Bring a pot of water to boil.  Add a handful of salt to season the water and add the spaghetti (I break mine in half to make easier to cook and eat).  Cook the pasta until it's al dente (I shoot for a minute shy of what the box says).  Drain and toss with 2 tablespoons of sesame oil, stir to coat and set aside.

Whisk the sauce ingredients (honey, peanut butter, soy sauce/tamari, rice vinegar, and sriracha) together or blend in a small blender or food processor (I prefer this method, gets a nice emulsification). 

In a large pot (like a dutch oven or soup pot, heat the pan on medium then add your remaining 1 tablespoon of sesame oil.  

Add the green onions (reserve 1/4 cup for garnish), carrots, napa cabbage, peppers, garlic, and ginger.
Saute for a minute or two until they have softened and barely start to turn color.  

Add back the chicken to the pan and then add the pasta and spaghetti squash. 

Add the sauce, tossing everything together to coat.  Taste to make sure the balance of flavors is right.  I often end up adding more peanut butter, tamari, salt, and/or sriracha.

Serve immediately, garnishing with the remaining green onions.