Place pork shoulder, half an onion, the halved head of garlic, and bay leaves in a large stockpot. Cover with water and bring to a boil. Reduce heat, skim any foam, and simmer for 2-2.5 hours until pork is tender. Remove pork to cool, then shred. Strain the broth through a sieve, discarding solids.

While pork cooks, toast the de-stemmed and de-seeded guajillo and ancho chiles in a dry skillet over medium heat for 30-60 seconds per side until fragrant. Transfer to a bowl, cover with hot water, and let soak for 20-30 minutes until soft.

Drain the softened chiles. Place them in a blender with 1/4 white onion, 3 cloves garlic, Mexican oregano, and 1 cup of the reserved pork broth. Blend until completely smooth. For an extra fine sauce, strain it through a sieve.

Return the strained broth to the pot. Stir in the red chile sauce, shredded pork, and rinsed hominy. Bring to a simmer and cook for at least 45-60 minutes to allow flavors to meld. Season with salt to taste.

Ladle the hot pozole into bowls. Serve immediately with desired garnishes like shredded cabbage, sliced radishes, diced onion, cilantro, and lime wedges.