In a large pot with boiling salted water, cook the spaghetti according to package directions. Add the broccoli florets the last 2 minutes of cooking and let them cook with pasta until tender. Drain the noodles and broccoli, and set aside.
While the pasta is cooking, whisk together the ginger, brown sugar, soy sauce, hoisin sauce, sesame oil, red pepper, and black pepper in a small bowl.
Season chicken with salt and pepper. Add olive oil to a medium-sized skillet and cook chicken in an even layer over medium heat, turning from time to time for 2 or 3 minutes, until slightly golden. Stir in the garlic, and carrots, and cook for a minute more, until cooked through but not overdone.
Transfer the cooked spaghetti and broccoli to the skillet, and pour the sauce on top and toss until fully incorporated. Reheat for a couple of minutes and serve the chicken stir fry noodles, garnished with green onions.