Start the soup: Heat oil in a large Dutch oven over medium. Add onion, carrots, celery, and garlic; cook, stirring often, until tender, 8 to 10 minutes. Stir in chicken broth, rotisserie chicken, salt, and pepper; cook, stirring often, until flavors meld, about 20 minutes.
Add the gnocchi: Add gnocchi and half-and-half, stirring to combine. Cook over medium, stirring occasionally, until gnocchi is hot throughout, about 6 minutes. Add spinach and sherry vinegar, stirring to wilt spinach. Divide evenly among 6 bowls, and garnish with basil. Serve immediately.
