Put the cooked beans and the bean cooking liquid in a blender and blend until totally creamy, scraping down the sides of the blender jar as necessary. The puree should have the consistency of pancake batter; add a bit of water or broth if needed to thin it out. Set aside.

Shape the sausage into 2 patties. Heat a generous glug of olive oil in a large Dutch oven or other large, deeper-than-a-skillet pot over medium heat. When the oil is hot, add the sausage and sear, smashing the patties down from time to time with your spatula to create a nicely browned crust on the bottom, about 5 minutes.

Flip the patties, add the garlic, and continue cooking until the sausage has browned and the garlic is nicely toasted and softened (but not too dark), another 4 to 5 minutes.

Add the carrot, celery, fennel, and onion and cook, stirring frequently, just until the vegetables begin to soften, 1 to 2 minutes. Add the pancetta and cook, breaking up the sausage and garlic into small bits with your spatula, until the pancetta has started to render some fat but isn’t yet crisp, 3 to 4 minutes.

Add the rosemary, chile flakes, and bay leaves. Add the bean puree and the Parmigiano rind, if using. Adjust the heat so the puree simmers gently. Simmer until the puree has reduced by about one-quarter and the other ingredients are nicely integrated, 15 to 20 minutes. Scrape the sides and bottom of the pot frequently to be sure the puree isn’t sticking anywhere.

Taste the puree; if it tastes flat (which it will if you’ve used canned beans), add a bit of salt, bearing in mind that you will be adding more salty ingredients when you finish the dish with salted pasta water and grated cheese. Then slide the pot off the heat.

Meanwhile, fill another large pot (at least 6 quarts/L) with 1 gallon (4 L) of water, add 4 tablespoons (40 g) kosher salt, cover the pot, and bring the water to a boil. If the water begins to boil before the bean puree is ready, turn down the heat, but don’t let the volume of the pasta water reduce by boiling off.

When the puree is ready, bring the pasta water (back) to a boil, add the noodles, and set your timer for 2 minutes less than the shortest suggested cooking time on the package of pasta; this will ideally be 2 minutes before the pasta is al dente. Stir the noodles several times during the first 2 minutes of cooking to prevent them from sticking to the bottom of the pot or otherwise clumping together.

When the timer goes off, start tasting the noodles. When they seem like they are 1 1/2 to 2 minutes away from a perfect al dente, drain and transfer them to the bean pot using your preferred method, making sure to reserve at least 1 cup (240 ml) of the pasta water.

Slide the bean pot back onto medium heat and finish cooking the noodles, tossing and adding plenty of splashes of pasta water until the noodles are perfectly al dente and the consistency of the sauce is creamy but not thick. If the sauce is too thick, add a little pasta water; if it seems watery, simmer for another few seconds to tighten it up, bearing in mind that the cheese will thicken it.

Reduce the heat to very low. Add the grated cheese and toss to emulsify it with the other sauce ingredients, adding splashes of pasta water (or plain hot water, if things are getting too salty) if needed to keep the consistency creamy and prevent the cheese from clumping. Given that the sauce is quite liquidy, you may not need to add any pasta water to encourage the cheese to melt.

Divide the pasta among warm bowls and serve right away, with more cheese to add at the table.