Toast the chiles, brown the meat. In a large (4-quart) saucepan or small Dutch oven, heat the lard, drippings or oil over medium. When hot, add the garlic and cook, stirring regularly until the cloves are golden. With a slotted spoon, scoop into a blender jar. Return the pan to the heat and, when hot, fry the chile pieces a small handful at a time, turning them nearly constantly with tongs or a slotted spoon as they toast and release their aroma. Immediately remove to a bowl, leaving behind as much fat as possible. When all are fried, cover with hot tap water, weight with a plate and rehydrate for 30 minutes.

Meanwhile, sprinkle the meat generously with salt and toss to coat evenly. Return the pan to medium-high heat and, if necessary, add more fat to coat the bottom of the pan. In uncrowded batches, brown the meat on all sides, transferring each batch to a rimmed baking sheet when ready.

Finish the sauce, braise the meat. Drain the chile-soaking liquid into a measuring cup. Taste: if the liquid is bitter, discard it. Scoop the soaked chiles into the blender jar and add 1¼ cup of soaking liquid or water. Add the oregano, pepper and cumin. Blend to a smooth puree, adding a bit more water if needed to keep the mixture moving through the blender.

Return the pan to medium-high heat and add more fat, if necessary, to coat the bottom of the pan. When hot, add the chile puree and stir continually until it has thickened to the consistency of tomato paste, 3 to 5 minutes. Return the meat to the pan, add the mushrooms and 1 ½ cups water, bring to a boil, reduce the heat to medium-low, partially cover and simmer until the meat is fork tender, about 1 hour. If the consistency is light and runny, simmer over medium to medium-high heat, stirring frequently, until the mixture easily holds its shape in a spoon.

Serve. Scoop the mixture into a serving bowl and set out with the queso fresco, onions and cilantro (if you’ve chosen to offer them) and warm tortillas for everyone to make tacos.