Place the chicken breasts in a saucepan and cover with water, then bring it to a boil. Reduce the heat, cover, and let it simmer gently for 10 to 12 minutes until it's cooked through and no longer pink inside. Once it's cool enough to handle, shred it into bite-size strips with two forks or your hands.
While the chicken cooks, whisk the peanut butter, soy sauce, garlic, chile-garlic sauce, and ginger together in a large bowl. I like to whisk it until it's completely smooth with no streaks of peanut butter left. If it feels too thick at this point, that's fine, since you'll thin it out later.
Bring a large pot of water to a boil and cook the spaghetti until it's just shy of tender, about a minute less than the package suggests. Add the vegetable medley to the same pot and cook for one more minute, just until everything is tender-crisp.
Drain the noodles and vegetables, but save a cup of the cooking water first. Rinse them briefly with cool water to stop the cooking. Stir a splash of the reserved cooking water into the peanut sauce to loosen it, then add the noodles, vegetables, and shredded chicken to the bowl and toss well until everything is coated.
Top the bowl with chopped peanuts and green onion right before serving. I like to taste it at this stage and add a little extra chile-garlic sauce if I want more heat, or a splash of soy sauce if it needs more salt.