Set a 6-quart Instant Pot to Sauté. Add the olive oil and Italian sausage and cook for 7–9 minutes, breaking it into bite-sized crumbles, until browned and cooked through. Drain excess fat if needed.
Add the onion and carrots and sauté for 3–4 minutes, until the onion begins to soften.
Add the garlic, tomato paste, Italian seasoning, fennel seeds, red pepper flakes, sea salt, and black pepper. Cook for 1 minute, stirring frequently.
Pour in 1 cup chicken bone broth and thoroughly scrape the bottom of the pot to release any browned bits.
Add the remaining 5 cups chicken bone broth and Parmesan rind, if using. Pour the crushed tomatoes over the top. Do not stir the tomatoes into the bottom of the pot to help prevent a burn warning.
Secure the lid and cook on HIGH pressure for 5 minutes. Once the cooking cycle is complete, carefully quick release the pressure.
Remove and discard the Parmesan rind. Set the Instant Pot to Sauté.
Stir in the refrigerated tortellini and cook uncovered according to the package directions, generally 3–5 minutes, or until just tender.
Turn off the Instant Pot. Stir in the coconut milk and spinach. Let sit for 2–3 minutes, until the spinach is wilted and the soup is creamy and warmed through.
Stir in 1 teaspoon balsamic vinegar. Taste and add the remaining teaspoon if desired. Adjust with additional sea salt and black pepper as needed.
Ladle into bowls and finish with freshly grated Parmesan, fresh basil, and additional red pepper flakes, if desired.