Preheat the oven to 350 degrees F and spread the raw walnuts on a baking sheet. Roast 3 to 5 minutes, or until walnuts are golden-brown. Allow walnuts to cool at least 5 minutes
Transfer walnuts to a food processor, along with the basil, cider vinegar, and sea salt. Pulse until ingredients are chopped. Turn the food processor on and slowly stream the oil through the hole in the top until combined. If desired, add more oil for a thinner consistency. Set aside until ready to use.
Cook pasta according to package instructions (note: I usually cook mine a few minutes less than is suggested on the package).
Heat a large non-stick skillet to medium and add the mushrooms and broccoli. Cook, stirring occasionally until mushrooms are golden-brown and soft, about 5 to 8 minutes.
Scoot the vegetables off to the side of the skillet and add the chopped chicken. Brown 2 to 3 minutes, then stir into the vegetables to combine.
Transfer all of the pesto sauce to the skillet and stir well. Cover and cook 5 to 8 minutes, until chicken has cooked through (note: reduce heat to medium-low if mixture is boiling out of control). Uncover and continue cooking until any excess moisture has burned off.
Transfer the cooked pasta noodles to the skillet with the chicken and veggies and carefully stir together (I use a spatula to fold everything together to avoid noodle breakage).
Taste the pasta for flavor and add sea salt (and/or parmesan cheese!) to taste.