Toast guajillo and ancho chiles in a dry skillet over medium heat for 20-30 seconds per side until fragrant. Transfer to a heat-safe bowl.
Pour hot water over toasted chiles and soak for 25-30 minutes until softened. Transfer chiles and 1 cup soaking liquid to a blender and puree until smooth.
Heat olive oil in a large pot over medium heat. Cook onion for 6-8 minutes until soft. Add garlic, tomato paste, cumin, and oregano; cook for 2 more minutes.
Add mushrooms and cook for 4-5 minutes until they release liquid and start to turn golden.
Strain chile puree into the pot and cook for 2 minutes. Add bay leaves and vegetable broth, bring to a boil, then reduce to a simmer.
Add drained hominy and beans to the simmering pot. Cook uncovered for 25-30 minutes, stirring occasionally.
Remove bay leaves, stir in lime juice, and season with salt and pepper to taste.
Ladle into bowls and serve with shredded cabbage, radishes, onion, cilantro, avocado, lime wedges, and oregano on the side. Enjoy with tostadas or tortillas.