Remove stems and seeds from the chiles and rinse well.
Place pork in a large stockpot and add just enough water to fully cover the meat. Bring to a boil, then reduce the heat to medium-low, and simmer until the pork is very tender (about 1 to 1.5 hours).
After the meat is cooked, add half of the chiles directly to the pot with the pork and let them soak until softened (about 20-30 minutes). Note: New Mexico chiles may require more time to soften.
Remove the chiles from the pot and peel off any large pieces of skin. Pulse the chiles in a blender or food processor, adding about 1 ½ cups of pork broth (from the pot) until the mixture forms a smooth puree.
Separately, blend the peppercorns, cumin seed (or ground cumin), and peeled garlic, adding ¼ cup of water to form a paste. You can use a traditional molcajete or spice grinder for the cumin and peppercorns, or substitute ground spices.
Remove the cooked pork from the pot, and chop it finely. Reserve the remaining broth to make the masa later.
Add 1-2 tablespoons of lard or oil to a stock pot, and saute the chopped pork over medium heat, gradually adding the spices and blended chiles to taste. Season with additional salt to taste.
Add pork broth as needed to make a moist, spoonable filling (“picadillo”).