Step 1

Toss cabbage, vinegar, and sugar in a medium bowl; season with salt and pepper and vigorously massage with your hands until cabbage is softened and slightly wilted.

Step 2

Do Ahead: Cabbage can be made 1 day ahead. Cover and chill.

Step 3

Cook avocados, cut side down, in a large skillet, preferably cast iron, over medium-high heat, until cut sides are blackened, 5–7 minutes. Transfer to a plate and let cool.

Step 4

Use a spoon to scoop avocados out of their skins, then cut each half lengthwise into quarters. Transfer to a plate and squeeze lime over top. Season with salt and sprinkle with chili powder.

Step 5

Season pork generously all over with salt and pepper. Heat oil in a large Dutch oven or other heavy pot over medium-high. Cook pork, turning occasionally, until browned all over, 15–18 minutes. Transfer to a cutting board and let cool.

Step 6

Reduce heat to medium and cook onions and garlic, cut side down, in the same pot, tossing onions occasionally, until onions are browned in spots and garlic is golden brown, about 5 minutes. Add cornmeal, chiles, and cumin seeds and cook, stirring constantly, until cornmeal is toasty smelling and cumin is fragrant, about 5 minutes. Add vinegar, stirring to release any bits stuck on bottom of pot, then add 8 cups water. Season generously with salt and pepper and bring to a simmer.

Step 7

Meanwhile, use a knife to slice pork meat from the bone, discarding any large pieces of fat. Cut meat into 2" pieces.

Step 8

Return pork meat and bone to pot and bring stew back up to a simmer. Cover with a lid, leaving slightly askew so steam can escape, and cook, adjusting heat to maintain a very gentle simmer and skimming foam from surface as needed, until meat is fork-tender, 1½–2 hours. Uncover pot and let stew cool (bone and all) until no longer steaming. Cover pot and chill stew at least 12 hours (you can skip this step, but it will dramatically improve the flavor).

Step 9

Uncover stew and spoon off half to three-quarters of fat on the surface; discard. Gently reheat stew until barely simmering. Pluck out and discard bone and garlic heads (don't worry about any cloves that may have escaped into stew). Taste and season with more salt and pepper if needed. Ladle into bowls and serve with Charred Avocados with Chili Powder, Tangy Red Cabbage, corn nuts, chicharrones, hot sauce, and/or sour cream as desired.

Step 10

Do Ahead: Stew can be made 3 days ahead. Keep chilled.