Step 1

At least 8 hours before cooking, sort the great northern beans, removing broken beans, stones, or dirt clods. Rinse the beans, put them in a large container, cover with 2 tablespoons Kosher salt and 4 quarts water, and stir to dissolve the salt. Let the beans soak for at least 8 hours, or overnight.

Step 2

Heat the olive oil in an Instant Pot with sauté mode set to high (or other pressure cooker set to medium-high heat) until the oil is shimmering. Add the onion, celery, carrots, red pepper flakes, and ½ teaspoon salt. Saute until the onions are softened and starting to brown, about 8 minutes.

Step 3

Rinse the great northern beans, drain, and add to the pressure cooker, along with the water and canned tomatoes. Tie the parsley stems and rosemary sprig into a bundle, and float on top of the liquid in the cooker, along with the Parmesan rind and the head of garlic. Lock the lid on the pressure cooker, and bring the cooker up to high pressure. Cook at high pressure for 18 minutes in an Instant Pot or other electric PC, or for 15 minutes in a stovetop PC. Let the pressure release naturally, about 20 minutes. (If you're in a hurry, you can quick release any remaining pressure after 15 minutes.) Remove the lid carefully, opening away from you - even when it's not under pressure, the steam in the cooker is very hot.

Step 4

Fish out the herb bundle, head of garlic, and Parmesan rind, and discard. Turn the heat to high under the cooker and bring the pot back to a simmer. Stir in the 1 tablespoon Kosher salt and the pasta. Simmer until the pasta is tender, about 9 minutes. (Note - do not lock the lid again - we aren't cooking the pasta under pressure.) Stir in the black pepper and balsamic vinegar, taste, and add more salt, pepper, and vinegar as needed.

Step 5

Ladle the soup into bowls, sprinkle the accompaniments on top, and serve.