Directions

• Score fat cap on pork shoulder in a 1 1/2-inch diamond pattern, cutting just until tip of the knife reaches meat. Flip pork shoulder over; cut along bone (about 3 inches deep) to open up meat. Rub pork shoulder all over with 3 tablespoons salt. Place on a rimmed baking sheet, and cover tightly with plastic wrap. Refrigerate 8 to 24 hours.
• Preheat oven to 375°F with oven rack in lower third of oven. Place pork shoulder, fat cap side up, in a large ovenproof Dutch oven. Add bay leaves and 6 cups water. Bring to a boil over medium-high. Cover and transfer Dutch oven to preheated oven. Roast until meat is fork-tender, 3 hours to 3 hours and 30 minutes, flipping pork shoulder after 2 hours. Let cool, covered, at room temperature 1 hour.
• Meanwhile, increase oven temperature to broil with oven rack in middle of oven. Place onion, tomatoes (cut sides down), and garlic on a lightly greased (with lard) rimmed baking sheet. Broil until onion and tomatoes are lightly charred and garlic is softened, 8 to 10 minutes for garlic and 10 to 14 minutes for tomatoes and onion. Set aside until cool enough to handle, about 10 minutes. Peel garlic cloves. Reduce oven temperature to 250°F.
• While tomato mixture broils, heat all dried chiles in a dry large cast-iron skillet over medium. Cook, turning occasionally, until chiles are fragrant and darkened in spots, 4 to 5 minutes. Transfer chiles to a small saucepan; add remaining 3 cups water, and bring to a boil over medium-high. Reduce heat to medium-low; cook, stirring occasionally, until chiles are softened, about 10 minutes. Drain, reserving chile cooking liquid.
• Combine onion, tomatoes, garlic, chiles, vinegar, sugar, oregano, remaining 2 teaspoons salt, and 1 1/4 cups reserved chile cooking liquid in a blender. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening. Process until smooth, about 30 seconds, adding up to 1/2 cup additional chile cooking liquid, if needed, to thin salsa to pourable consistency. Season with salt to taste.
• Remove pork shoulder from Dutch oven; discard braising liquid. Pull meat into bite-size pieces; discard bone and large pieces of fat. Heat 2 tablespoons lard in cast-iron skillet over medium-high. Add just enough pork (2 to 3 cups) to cover bottom of skillet in a single layer, and press lightly to flatten. Cook until crispy and lightly browned, 4 to 6 minutes, flipping meat halfway through cook time and seasoning both sides with salt to taste while cooking. Transfer pork to an ovenproof serving platter, and keep warm in 250°F oven. Repeat with remaining pork, adding more lard to skillet as needed. Serve with corn tortillas, shredded cabbage, cilantro leaves, lime wedges, and salsa.