Set a large 6-8 quart soup pot (or Dutch oven) over medium heat. Add the olive oil to the pot. Once hot, stir in the onions, celery, carrots, and garlic. Stir and sauté for 2 to 3 minutes.
Stir in the all-purpose flour and Italian seasoning. Sauté another 1-2 minutes to cook the flour. Then stir in the chicken base, and water.
Cover the pot and bring to a boil. Simmer the soup for 10 to 15 minutes, until the vegetables are tender. (*If adding raw chicken breasts to the soup, make sure to simmer at least 15 minutes, but up to 20 minutes. Then remove the chicken breast to shred before adding the gnocchi.)
Remove the lid and stir in the gnocchi. Simmer for 2-4 minutes until the gnocchi starts to rise to the top of the soup. Then mix in the Parmesan cheese, half-and-half, and chopped chicken. Stir again to incorporate the cheese into the soup.
Finally, stir in the fresh basil leaves. Taste, then salt and pepper as desired. If you wish the soup to be slightly thicker, allow it to simmer another 3 to 5 minutes. Serve warm.
