In a large soup pot or Dutch oven heat the butter and oil together over medium heat. Add onion, carrots, celery and garlic. Season with salt and pepper and cook until tender over medium-low heat about 10 minutes, stirring often.
Sprinkle the flour into the pot and stir into the vegetables, cook 3 minutes, stirring often. Stir in the chicken broth 1 cup at a time, followed by the half & half.
Add the cooked chicken. Bring up to a simmer and maintain for 20 minutes stirring often. Season with salt and pepper to taste.
Cook the gnocchi separately according to package directions. Add them to the soup along with the spinach. Simmer until spinach is wilted. Stir in basil. Serve with grated cheese.