Heat oil in a large Dutch oven over medium. Add sausage; cook, stirring often to break into small chunks, until browned, about 6 minutes. Using a slotted spoon, transfer sausage to a plate; leave oil in Dutch oven.

Add onion, carrot, celery, salt, and black pepper; cook over medium-high, stirring often, until softened, about 8 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute. Add vinegar; cook, stirring to scrape up browned bits, until completely reduced, about 30 seconds.

Sprinkle in flour and cook, stirring constantly, until flour is lightly toasted, about 1 minute. Gradually stir in stock until well incorporated. Add Parmigiano-Reggiano rind and thyme sprigs. Bring to a boil over medium-high. Reduce heat to medium-low, and simmer, uncovered, stirring occasionally, until flavors meld and vegetables are tender, 12 to 15 minutes. Remove rind and thyme sprigs, discarding the sprigs but reserving the softened rind for a snack or chopping it and adding it back to the soup if desired.

Add tortellini and cooked sausage; simmer, stirring occasionally, until tortellini is tender, 2 to 3 minutes. Reduce heat to low. Stir in cream and spinach; cook, stirring occasionally, until spinach is wilted, about 2 minutes. Divide soup evenly among 6 bowls. Garnish each with 1 tablespoon Parmigiano-Reggiano and thyme leaves. Serve immediately.