Roast the unpeeled garlic on an ungreased griddle or heavy skillet over medium heat, turning occasionally, until soft (they will blacken in spots), about 15 minutes; cool and peel. While the garlic is roasting, toast the chiles on another side of the griddle or skillet: 1 or 2 at a time, open them out flat and press down firmly on the hot surface with a spatula; in a few seconds, when they crackle, even send up a wisp of smoke, flip them and press down to toast the other side. In a small bowl, cover the chiles with hot water and let rehydrate 30 minutes, stirring frequently to insure even soaking. Drain.
In a food processor or blender, combine the chiles and a little water, garlic, oregano, pepper and cumin. Blend to a smooth puree, scraping down and stirring frequently. (If the mixture won’t move through the blender blades, stir in a tablespoon or two of water to get things going.) With a rubber spatula, work the chile mixture through a medium-mesh strainer into a bowl.
Heat the oil in a medium-size (4-quart) pot (preferably a Dutch oven or Mexican cazuela) over medium-high heat. When hot, add the lamb and brown thoroughly, about 8 minutes. Add the chile puree and stir for about 5 minutes as the mixture thickens and concentrates all those rich, pungent flavors. Stir in broth and simmer until lamb is tender, about 2 hours. Add mushrooms, epazote (or cilantro), Tabasco sauce and masa harina. Partially cover and simmer gently over medium-low, stirring frequently, until mushrooms are soft and the sauce has reduced to coat the meat rather thickly, about 30 minutes. Taste and season with salt. Serve with goat cheese, chives, tortilla strips, cilantro and more hot sauce for each person to add to the chili as they want.