Posole is a staple in these parts. Mexican soup. And like chicken noodle or any other soup, it has a million variations. You need hominy, probably pork but chicken isn't verboten, a clearish non-tomato-y but still thick broth but tomatoes aren't unheard of, chiles of any flavor, spices ... and the rest is left up to you.
You can cook it in the crock pot or on the stove. You can cook the meat in the liquid or brown it first on the stove or braise it in the oven. There's basically no wrong way to cook it, as long as you end up sort of in the same space.
Your building blocks are:
Meat: 1.5-2 lb pork shoulder, boston butt, picnic, or any roast. Pork loin or tenderloin is not unheard of.
Hominy: either dried or canned. Dried will require overnight soaking and then boiling. Canned is fine. About 4 cups.
One onion, a few garlic cloves, chiles of any sort (sweet or hot).
Chiles: Yeah, I know I already said chiles, but wanted to make sure. Freshly roasted is great, canned are alright, frozen are convenient if you can get them in your neck of the woods. About a cup, more or less to taste. Mild or hot, doesn't matter.
Spices: Cumin, oregano, cayenne, maybe cloves. I've seen folks prep the meat in a stock with cloved onions, whole peppercorns, and cumin seeds, and then use the ground / powdered later as well. Pre-mixed chili powder is cool as well. I'm digging Savory Spices' no-salt chili powder right now, a lovely mix of Ancho and Chimayo chile, garlic and cumin, paprika and cayenne, and Mexican oregano.
Step One: Prep the meat.
There are three basic ways to get started. You can chop the meat up, add it to the soup while it's cooking, and cook the meat in the soup liquid. That's the easy way, because it's fewer steps, but the meat gets a little bland and everything can taste watered down. A very common way to get started is to simmer the meat in salted water or broth (chicken or veggie stock). If you're using water, maybe add a clove-studded onion, garlic cloves, peppercorns, cumin seeds, and even a carrot or celery stick or two.
Yet another way, and the method pictured here, is to brown the pork shoulder first. Pre-heat your oven to 350º. Then prep the meat by chopping your pork shoulder in several 3-4 inch wide chunks. If you're using a bone-in shoulder, leave some meat on the bone. Using an oven-safe pan with a lid (a Dutch oven would have worked as well), take a couple of onions, sliced fairly thin, and sauté over medium-high heat in olive oil, 3-4 minutes, and then add a mix of chili powder and Mexican oregano (one Tbs each), some freshly ground pepper, and cayenne or red chili pepper flakes, if you're into that.
Then add the pork chunks and brown on all sides, just 2 minutes tops per side. At this point I threw in some carrot and celery chunks, totally optional. Add 4-5 cups water or broth, bring to a boil, cover and transfer to the oven. Braise for 1-2 hours, depending on how much meat you have. The meat is ready in one hour, while two hours will give you a richer stock.
When it's done, you're going to remove the meat and set it aside, then strain out the onions (and carrots and celery, if you used it), and use the stock for the soup. After the meat has cooled, chop it or shred it into bite-sized pieces.
Since you have the oven already going, you can roast a couple of jalapeños at the same time. I use a gringo technique of slicing them in half, removing the seeds, then wrapping in tinfoil and baking for 30 minutes or so, just enough to soften them up and slightly sweeten them. When they have cooled, chop them up and set aside.
In a large stock pot (or, again, a Dutch oven works here as well), sauté another 1-2 onions in olive oil, medium-high, 3-4 minutes. Add a few cloves worth of chopped garlic. Add about 2-3 Tbs total a mix of cumin, chili powder, cayenne, black pepper, and/or Mexican oregano, and stir to coat the onions evenly. Then add the shredded pork and continue stirring for a few minutes to again evenly distribute the spices.
Add hominy, stir, and then pour in the reserved pork broth. (You can always top it off with chicken broth, if you don't have enough to cover the pork and hominy.) Then add about a cup of chopped green chiles and the oven-roasted jalapeños.
Simmer for at least 45 minutes, until the hominy is tender. Simmer for up to 2 hours to maximize the flavor of the broth, to let it soak up the chile and pork flavors.
Depending on how well you strained your pork broth, you might need to skim a little fat off the top of the soup before serving. Or, if it's winter time, leave it there for extra warmth!
You can serve it up as is, or you can add a little bit of chopped green cabbage, white onion, and chopped cilantro. A squirt of lime juice is not unheard of. Warm tortillas on the side, of course.