In large pot of water, cook onions for 3 minutes. Remove with slotted spoon; set aside. In same water cook pasta according to package directions. Reserve 1 cup cooking water. Drain pasta; set aside.
While pasta cooks, in 12-inch skillet cook chicken in hot oil over medium-high heat for 3 to 4 minutes, stirring occasionally. Add broccolini and onions. Sprinkle parsley, salt, and red pepper. Cook, stirring frequently, for 5 minutes. Add tomatoes, broth, reserved water, and spaghetti. Cover; simmer about 5 minutes or until broth is slightly reduced and vegetables are tender.