Rinse and pick over beans; place in a large bowl, add 3 teaspoons salt, and cover generously with water. Set aside to soak for 12 to 24 hours.

Put ham shank, onion, celery, carrot, garlic, peppercorns, bay leaves, rosemary, and parsley in the liner of an electric pressure cooker. Add 3 quarts of water, seal, and set to cook under high pressure for 1 hour. When time is up, turn the pressure cooker off and let vent naturally for 20 minutes; carefully remove lid. Strain stock; reserve ham shank and discard remaining solids. When shank is cool enough to handle, pick off meat and set aside; discard skin, fat, gristle, and bone.

Return the stock to the pressure cooker; drain beans and add to stock. Seal and set to cook under high pressure for 3 minutes, then carefully quick-release the pressure. Drain the beans, reserving the stock; set both aside.

Meanwhile, make the base:

Heat 2 tablespoons of olive oil in a Dutch oven over medium heat. Add pancetta and cook, stirring occasionally, for 10 minutes; adjust heat so it does not brown. Add onion, carrot, celery, garlic, and salt to taste and cook, stirring occasionally, until vegetables are very soft, about 15 minutes. Increase heat; when the vegetables begin to brown, add garlic and cook until fragrant, 1 minute. Make a well in the center and add a splash of olive oil and the tomato paste; cook, stirring, until the tomato paste caramelizes a bit and is very fragrant, 3 to 4 minutes. Stir the pan through and add the tomatoes, Parmesan rind if using, rosemary, and half the reserved stock. Bring to a simmer, then taste and adjust salt and pepper as desired; add the cooked beans and simmer for 30 minutes while adding the chopped vegetables, longest-cooking first, greens at the last minute.

Meanwhile, bring the remaining stock to a boil and add the pasta. Cook, stirring occasionally, for a few minutes until pasta is just cooked; drain, reserving cooking liquid.

If minestrone is thicker than desired, add some of the reserved pasta cooking liquid to the pot to reach desired consistency (some people like it really thick, some people like it more brothy). Taste minestrone and add a splash of red wine vinegar if desired.

To serve, divide pasta among wide soup bowls and top with minestrone. Drizzle each bowl with a little olive oil and grate cheese over the top.

Note: Soaking the beans in salted water will toughen the skins, and the beans will hold up in the finished soup. If you prefer a thicker finished soup, eliminate the salt in the soaking water.