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Preheat the oven to 350 degrees F. Place the chicken breasts on a sheet pan. Place the veggies on the same pan. Rub the tops with olive oil and sprinkle liberally with salt and pepper. Roast for 35 minutes, until cooked through. Set aside until cool enough to handle. Shred the chicken. Set the veggies aside.
Place the garlic, ginger, canola oil, tahini, peanut butter, soy sauce, sherry, sherry vinegar, honey, chili oil, sesame oil, 1/2 teaspoon black pepper and the cayenne pepper in a food processor fitted with a steel blade and puree until smooth. Pour the sauce into a large serving bowl.
Boil a large pot of salted water and cook the spaghetti according to the directions on the package. Drain the pasta in a colander, place it in the bowl with the sauce. Add the peppers, chicken and broccoli and toss well. Top with cashews and sesame seeds. If desired serve with extra hot oil.