Pat the pork dry with paper towels, then season it with 1 teaspoon of the kosher salt and the black pepper. If using fresh chiles, roast, steam, peel, seed, and chop them before beginning the pork so the cooking process stays organized.

Heat the oil in a large Dutch oven or heavy pot over medium-high heat. Brown the pork in batches for about 2-3 minutes per side, then transfer each batch to a plate so the pieces do not steam.

Reduce the heat to medium and add the diced onion. Cook for 5-7 minutes, stirring occasionally, until softened, then add the garlic, cumin, Mexican oregano, and optional cayenne and cook for about 30 seconds.

Add the chopped roasted green chiles and optional tomatillos to the pot. Cook for 3-5 minutes, stirring occasionally, until the vegetables begin to soften and their flavors mingle with the seasoned onion.

Return the browned pork and any accumulated juices to the pot. Pour in the chicken stock and add the remaining 1/2 teaspoon salt; the liquid should come about halfway up the meat.

Bring the mixture to a simmer, partially cover the pot, and cook over low heat for 1 1/2-2 hours. Stir occasionally, keeping the heat gentle, until the pork is fork-tender and easily breaks apart.

Remove the lid during the final 15-20 minutes if the chili needs to thicken. Stir in the chopped cilantro and lime juice, taste carefully, and adjust the salt before serving hot.