1. Drain and rinse the hominy, place it in a large pot and add 3 liters of vegetable broth. As soon as it boils, lower the temperature and simmer, covered, for about 1 hour.
2. Clean the mushrooms with a basting brush and remove any remaining impurities with a damp cloth. Pull the cleaned mushrooms into strips.
3. Sear the mushrooms with some oil in a very hot frying pan. Be sure to do this a little at a time so that the mushrooms are well browned. The garlic cloves can also be roasted at this step. Set aside.
4. Clean the chiles, removing seeds and stem.
5. Soak the chiles in some of the broth until softened (5 to 10 minutes).
6. Now puree them together with the roasted garlic, onion, oregano and some broth in which the chilies were soaked. Strain to obtain a smooth paste.
7. Heat a bit of oil in a small saucepan and sauté the strained chili sauce. Season with salt and let simmer until it has thickened and changed color (about 10 minutes). If it is too thick, add some of the hot broth.
8. Stir the resulting sauce into the boiling broth. Be careful not to add it all at once, but gradually, so that you can season the broth to taste.
9. Let the broth simmer until the corn softens and the kernels pop. This process can take another 30 to 40 minutes. Add a little boiling water if necessary
10. Add the seared mushrooms and let cook for another 10 minutes.
11. Meanwhile, finely chop lettuce, onion and radishes and set aside to serve.
12. Season the pozole with salt and serve with freshly ground piquín chile, lettuce, white onions, radishes, Mexican oregano and quartered lemons. Hand tostadas (crispy baked or deep-fried corn tortillas) with it and enjoy.