1 teaspoon mild chile flakes, such as Aleppo, plus more for serving
2 fresh or dried bay leaves
3 tablespoons tomato paste (double concentrate)
1 dried shiitake
8 ounces dried medium-size white beans, such as Corona, baby lima (butter beans), gigante, or cannellini, preferably soaked overnight
1 or 2 Parmigiano cheese rinds (optional), plus Parmigiano for grating
1 large bunch Tuscan, green, or red kale, stems removed, leaves torn into 2- to 3-inch pieces
8 ounces small pasta, such as orecchiette, shells, or ditalini
Extra-virgin olive oil and crusty bread, for serving
6 garlic cloves
2 yellow onions, halved
4 carrots, scrubbed and roughly chopped
1 fennel bulb, stems trimmed, roughly chopped
1/3 cup extra-virgin olive oil, plus more for serving
Kosher salt and freshly ground black pepper
Heat a 6-quart Dutch oven or other heavy pot with a lid over medium. If you have just finished the soffritto, continue in the same pot. If not, add soffritto to pot and cook until sizzling. Recipe for soffritto below.
Add the anchovies, if using, along with oregano, chile flakes, and bay leaves. Cook, stirring, until anchovies have disintegrated, 1 to 2 minutes.
Add tomato paste and stir to coat; cook until paste has darkened by a couple of shades, about 3 minutes more.
Add shiitake and then beans, along with any soaking liquid. Beans should be covered by 2 or 3 inches of water; add more from the tap if needed. Drop Parmigiano rind into liquid, if using.
Bring mixture to a boil over medium-high, season generously with salt and pepper, then reduce heat to maintain a gentle simmer. Cook, stirring every 15 minutes or so, until beans are starting to soften, 1 to 2 hours depending on whether your beans were soaked (add more water as needed to keep beans submerged by 2 inches). The best way to check is to taste a bean; the center can be firm but without any crunchiness or chalkiness, and the outer edge of the bean should feel pretty soft.
Add kale and stir to wilt and submerge the greens in liquid. Cook soup with lid askew, stirring occasionally, until beans are completely soft but still hold their shape and kale is silky, about 1 hour more. Taste periodically and season as needed.
When the beans are just about done cooking, cook pasta in a pot of well-salted boiling water for 2 to 3 minutes less than the time indicated on package (it will finish cooking in the soup). Drain pasta and add to soup, then taste and adjust seasoning. (Do not try to skip a step by cooking the pasta in the soup. The noodles will absorb all the liquid and the liquid will be thick and gummy.)
Serve soup topped with a drizzle of olive oil and with Parmigiano for grating over, bread for dunking, and extra chile flakes.
Combine garlic and onions in a food processor, then pulse until vegetables are finely chopped. Transfer to a large bowl.
Add carrots to processor and pulse until finely chopped; add to bowl with onions and garlic. Repeat with fennel, combining with other vegetables when done.
Heat a 6-quart Dutch oven or other large heavy pot with a lid over medium. Add olive oil and cook until oil is hot, about 1 minute, then add chopped vegetables. Season mixture generously with salt and pepper, stir to combine, and cook until vegetables start to release some liquid, about 3 minutes.
Cover pot and cook soffritto, checking and stirring every 5 to 10 minutes, until vegetables are very soft but have not taken on any color, about 15 minutes. Mixture will still look bright and there will be lots of bubbling liquid from the vegetables.
Continue to cook, stirring every 5 minutes or so, until soffritto is just starting to brown and has lost at least half its volume, about 10 minutes more.
The soffritto is done when it has darkened by several shades and vegetables are starting to fry in the oil, which indicates that they’ve released all of their liquid. You should be able to smash the vegetables to mush by pressing on them with the back of a wooden spoon. This could take 35 to 45 minutes in total.