Thaw shrimp, if frozen. Peel and devein shrimp, leaving tails intact. In a 4 quart Dutch oven bring a large amount of water to boiling. Add the spaghetti; cook 4 minutes. Add the broccoli; cook 2 minutes; Add the shrimp; cook 2 to 3 minutes more or until pink.
Meanwhile, in a bowl combine peanut butter and soy sauce. Stir in vinegar, sesame oil, chili oil, gingerroot, and garlic. Drain pasta mixture; return to pan. Add peanut butter mixture, onions, and nuts; toss gently.