In a Dutch over or similar large soup pot, heat 1 tablespoon of the oil over medium-high heat. Add the beef and 1/2 teaspoon of the salt. Cook, breaking apart the meat with a wooden spoon or spatula, until no pink remains, 5 to 7 minutes. Remove to a plate.
Reduce the heat to medium. Add the remaining 1 tablespoon oil, carrots, onion, and remaining 1 teaspoon salt. Cook, stirring frequently, until the vegetables soften and the onion begins to turn translucent, 8 to 10 minutes. Add the garlic and cook until fragrant, about 30 seconds. Add 1 cup of the chicken broth. Bring to a simmer and scrape the spoon along the bottom of the pan to remove any stuck-on bits of food.
Add the crushed tomatoes and increase the heat to medium high. Let the tomatoes simmer for 5 minutes, stirring and scraping the spoon along the bottom of the pan often to keep them from sticking.
Add the remaining 3 cups broth, kidney beans, Great Northern beans, bayleaf, Italian seasoning, pepper, red pepper flakes, and Parmesan rind (if using). Bring to a gentle simmer. Let cook, stirring every few minutes, for 10 minutes.
Carefully transfer a few ladlefuls of the soup to a blender, filling the blender no more than halfway (be careful not to scoop up the bayleaf of Parm rind). Puree until smooth, then stir back into the soup (you also can place an immersion blender directly in the pot and blend BRIEFLY. You want to puree part of the beans to thicken the soup but still have plenty of beans intact—remove the bayleaf and Parmesan rind while you do this to make sure you don't blend them, then add them back to the soup).
Stir in the pasta. Bring the soup to a constant, rapid simmer and let cook, stirring often to keep the pasta from sticking to the bottom of the pot, until the pasta is al dente, about 15 minutes. Partially cover the pot to keep the soup from splattering all over your stove.
Stir in the red wine vinegar, parsley, and cooked beef. If the soup is thicker than you would like, add water or additional broth, until your desired consistency is reached. Taste and adjust seasoning as desired (I usually add a few pinches of salt). Serve hot, with a generous sprinkle of Parmesan, a drizzle of olive oil, and additional fresh herbs.
Notes
Adapted from the Instant Pot Pasta e Fagioli in my cookbook.
TO STORE: Store leftover pasta e fagioli in an airtight container in the refrigerator for up to 5 days.
TO REHEAT: Warm on the stovetop or in the microwave until steaming.
TO FREEZE: Freeze pasta e fagioli for up to 3 months. Thaw overnight in the refrigerator before reheating.