Preheat the oven to 350 degrees F. Place the chicken breasts on a sheet pan, skin side up. 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. Remove the meat from the bones and discard the skin and bones. Shred the chicken in large bite-sized pieces and set aside.

Meanwhile, bring a large pot of salted water to a boil and blanch the broccoli florets for 2 minutes. Drain and immediately immerse in a bowl of ice water to stop the cooking and set the color. Drain and set aside.

Place the garlic, ginger, vegetable 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. Set aside.

Add a splash of olive oil to a large pot of boiling salted water and cook the spaghetti according to the directions on the package. Drain the pasta in a colander, place it in a large bowl, and while still warm, toss with the sauce. Add the scallions, peppers, chicken and broccoli and toss well. Serve warm or at room temperature.