Rinse and drain the hominy and beans. In a large pot over medium-high heat, warm 2 tablespoons vegetable oil. Add the mushrooms and sauté until they release their moisture, 5 minutes. Add 1 cup of the hominy, the chili base, ⅔ of the beans, and 2 cups beer, broth, or water. Add a bay leaf, if you have one. Bring to a simmer and cook until thickened slightly and the flavors have combined, 10 minutes. Stir in the juice of ½ lime. Season with salt to taste.
Meanwhile, prepare the toppings: Slice a couple of radishes. Pick the cilantro leaves and toss the stems. Cut the remaining ½ lime into wedges for squeezing. Set out the shredded cabbage, tortilla chips, and jack.