Food Network teaches how to crush, slice and mince garlic. Peel off some of the papery skin from the garlic and then smash the head of garlic with the heel of your hand to loosen the cloves; if you only need a few cloves, leave the head intact and pull some off. Separate the cloves. To peel a clove, cut away the root end with your knife. Lay the flat side of the knife over the clove while holding the knife handle, then with the heel of your free hand carefully whack the knife against the garlic to separate the skin from the clove. To crush the peeled garlic, lay the flat side of the knife over the clove and smash it again. To slice peeled garlic, lay the clove flat on the cutting board and hold it with the fingertips of one hand, keeping them curled under. Using a rocking motion with the knife, make thin slices by moving the knife slowly across the clove. To mince peeled garlic, lay the flat side of a knife over the clove and smash it. Roughly chop the clove then move your free hand flat across the tip of the knife and use a rocking motion to chop the garlic until it’s finely minced.
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.