Add the oil to a 6-quart or larger Instant Pot and select "saute." Once the oil is hot, add the sausage. Cook, breaking the meat into small pieces, until no pink remains, 5 to 7 minutes.
Add the carrots, onion and salt. Cook for 4 minutes, stirring frequently, until the onion begins to soften and turn translucent. Add the garlic and cook until fragrant, about 1 minute. Carefully splash in about ½ cup of the broth. Scrape a spoon or spatula along the bottom of the pot and remove any stuck-on bits.
Add the remaining broth, tomato sauce, diced tomatoes, kidney beans, great Northern beans, pasta, Italian seasoning, black pepper, red pepper flakes and Parmesan rind (if using). Stir to combine.
Close and seal the Instant Pot. Cook on high (manual) pressure for 3 minutes. Allow the pressure to release naturally for 10 minutes, then immediately vent to remove any remaining pressure. Carefully open the lid. Remove the Parmesan rind. Stir in the vinegar. Taste and adjust the seasoning as desired. Serve hot, sprinkled with Parmesan and fresh basil.
Notes
Nutritional information does not include Parmesan sprinkled on top.
Freeze your Parmesan rinds in a zip baggie whenever you have them, and then you'll have them available to add to soup!
Storage tips: Refrigerate leftovers for up to 5 days, or freeze for up to 5 months. Let thaw overnight in the refrigerator. Rewarm gently in the microwave or on the stovetop.