Step 1
Melt 4 Tbsp. duck fat or unsalted butter in a large Dutch oven or other heavy pot over medium-high heat. Add 2 medium onions, quartered through root ends, 4 medium leeks, white and pale green parts only, halved lengthwise, thinly sliced, 6 medium carrots, peeled, quartered lengthwise, cut into 2″–3″ pieces, 8 garlic cloves, halved, and 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt; season with freshly ground pepper. Cook, stirring occasionally, until carrots are slightly softened and onions are translucent, 5–7 minutes.
Step 2
Pour in ¼ cup cognac or whiskey and cook, stirring, until evaporated, about 1 minute. Make a well in the center of onion mixture and add 2 lb. pork shanks with skin, soaked overnight in fridge, drained, and 2 lb. pork neck bones and/or pig’s tails and/or ears (if using). Arrange 1 small head of savoy cabbage, cored, cut into 8 wedges, in a mostly even layer in pot; sprinkle with 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt. Pour in 7 cups water and bring to a boil, skimming any foam from surface as needed. Reduce heat, cover pot, and cook, adjusting heat as needed to maintain a simmer, 1½ hours.
Step 3
Stir in 1½ cups white beans, soaked overnight, drained, cover pot, and cook, stirring occasionally and skimming any foam from surface, 1 hour. Add 8 small Yukon Gold potatoes, peeled, pushing down into soup to submerge; add up to another 1 cup water if needed to cover. Return soup to a simmer and cook, uncovered, stirring occasionally, until beans and potatoes are tender, 60–75 minutes.
Step 4
Using tongs, transfer pork shanks and neck bones to a
cutting board and let cool slightly. Pick meat from bones; discard skin, bones, and
large pieces of fat. Shred meat and return to pot. Taste soup and season with more salt
and pepper if needed.
Step 5
Ladle soup into bowls. Serve with toasted country-style bread and room-temperature butter alongside.
Do Ahead: Soup can be made 3 days ahead. Let cool; cover and chill. Reheat, covered, over medium-low heat.