This is not "authentic" but a short-cut recipe but that does not mean that it is not good.
Cut up the leftover pork shoulder and clean it off of the bones as well as you can. Put the pork and the bones in a large pot and add a chopped onion and 2-3 cloves minced garlic. Cover with cold water and bring to a boil and then reduce the heat to a simmer, skim if necessary. Add about a heaping 1.5 tsp of Mexican oregano, crumbled. If I am being labor intensive and have time I take several dried chilies, anchos & pasillas and stem and seed. Open flat and toast on a griddle or skillet, careful they can have you sneezing and coughing quickly. Then place in a deep bowl and add cold water to cover. Let them soak about 20 minutes then place in a blender with a little of the soaking water and blend. Strain into the pot of posole. If you are pressed for time then add good pure chili powder, ground anchos, pasillas, New Mexicos etc (available from www.penderys.com) about a Tbs or 2 and simmer some more. Add a drained 1 lb.can of Juanitas hominy, I like the texture better.Simmer some more, chill a beer and eat plain or with a squeeze of lime, some shredded cabbage, some diced avocado..
I adjust all of this by the amount of pork that I have and am sorry I cannot be more specific about measurements, it is just something that I look at as i go along and say okay that is enough of that or I need some more of this. I get my Mexican oregano from Pendery's too, not Penzy's.