Heat a large pot over medium. Add oil, then onion, carrot, seasoning and salt. Season with pepper.
Cook, stirring occasionally, until onion is soft, 4 to 6 min. Add tomato paste and turkey.
Cook, breaking up meat with a wooden spoon, until meat is no longer pink, 4 to 5 min.
Stir in beans and liquid, 4 cups water and pasta. Bring to a boil over medium-high and cook, stirring occasionally, until pasta is tender, about 10 min.
Stir in spinach and cook until just wilted, about 1 min.
Divide minestrone among bowls and sprinkle with cheese.