Heat the oil in a heavy large frying pan over medium heat. Add the onion and garlic and saute until translucent, about 5 minutes. Add the carrot and celery and saute until the vegetables are tender, about 5 minutes. Add chicken and and heat 1 minute. Add the marinara sauce. Decrease the heat to medium-low and simmer gently for 15 minutes to allow flavors to blend, stirring often. Stir in basil. Season the sauce, to taste with salt and pepper. (The sauce can be made 1 week ahead. Cool completely, than transfer it to a container and freeze for future use. Bring sauce to a simmer before using.).
Meanwhile, cook the spaghetti in a large pot of boiling salted water until just tender but still firm to bite, stirring often, about 8 minutes. Drain, reserving 1 cup of the cooking liquid. Add the pasta to the sauce and toss to coat, adding enough reserved cooking liquid to moisten as needed. Serve with Parmesan.