Trim pork shoulder of excess fat and sinew. Cut into 1-inch pieces (you should have about 3 pounds cubed pork after trimming). Season meat with 1 tablespoon kosher salt and 1 teaspoon freshly ground black pepper.
Heat 2 tablespoons oil in large heavy-bottomed pot or Dutch oven (7–8 quart) over medium-high heat until shimmering. Add half of pork and sear until browned on all sides, about 5 minutes. Transfer pork to a bowl and repeat with remaining pork.
Return all meat back to pot, add 12 cups water and bring to a simmer, skimming off most of foam that rises to surface. Add 3 fresh bay leaves and 2 tablespoons oregano and return to a gentle simmer while you make chile purée.
Heat remaining 1 tablespoon canola oil in a separate medium (3-4 quart) heavy-duty pot over medium heat. Add 2 ounces dried chiles and toast over medium heat, stirring often, until blistered and beginning to blacken in spots, about 2 minutes. Add 3 cups diced onions and 4 cloves smashed garlic and sauté until fragrant and onions begin to soften, about 2–3 minutes. Ladle 3 cups simmering pork broth over top of chiles; cover pot and simmer until chiles and onions are tender, about 15 minutes.
Remove pot with chiles from heat and let cool for about 10 minutes. Transfer chile mixture (with its liquid) to a high-powered blender with 1 chipotle and 1 tablespoon adobo sauce. Blend until very smooth, about 1 minute.
Stir chile purée, 2 (25-ounce) cans drained hominy and 1 teaspoon salt into pot with simmering pork. Continue gently simmering until broth is rich and hominy and pork are tender, about 1 ½ hours.
Remove bay leaves and stir in 2 tablespoons lime juice and 1/4 cup chopped cilantro. Taste and season with additional salt and lime juice as needed.
Ladle pozole into wide bowls and garnish generously with your choice of toppings and a crunchy tostada broken over top.