Heat the olive oil in a large pot over medium heat until shimmering, about 1 minute.
Add the Italian sausage and cook, breaking it into small pieces, until browned and no longer pink, 6–8 minutes. Spoon off excess fat if needed.
Stir in the onion, carrot, and celery; cook, stirring occasionally, until softened and lightly golden, 5–7 minutes.
Add the garlic, tomato paste, and red pepper flakes; cook, stirring, until fragrant and the paste darkens slightly, 1–2 minutes.
Pour in the crushed tomatoes and chicken broth, then add the bay leaf, dried oregano, Parmesan rind, and kosher salt. Bring to a boil, reduce to a lively simmer, and cook uncovered until flavors meld, 15–20 minutes.
Stir in the cheese tortellini and simmer until they float and are just tender, 4–6 minutes (check package for timing).
Add the baby spinach and cook, stirring, until just wilted, 1–2 minutes. Remove and discard the bay leaf and Parmesan rind.
Off the heat, season with the black pepper and adjust seasoning to taste. Ladle into warm bowls and top with Parmigiano Reggiano for serving.