Cut the pork into bite-size pieces. Season the pork chunks generously with salt and pepper and set aside.
Remove and discard the stems and seeds from the dried chiles. Tear or roughly chop the chile peppers into medium-size pieces, roughly ½-inch wide.
Toast the torn chiles in a skillet over medium heat while stirring for 1-2 minutes until fragrant. Transfer to a medium pot and fill with enough of the 2-3 cups chicken broth or water to cover the chiles.
Bring it to a boil, then cover and remove it from the heat. Let the chiles soak for 10 minutes. Do not discard the cooking liquid - you will need 2 cups of this liquid for the sauce.
Transfer the soaked chiles to a blender using a slotted spoon. Add 1 cup of the chile-infused water along with 4 garlic cloves, the chipotle peppers in adobo sauce, vinegar, oregano, thyme, brown sugar, ground cumin, and salt. Blend well until smooth. Add an additional 1 cup of the chile liquid and blend.
Heat 2 Tablespoons of oil in a large Dutch oven or heavy-duty oven-safe pan over medium-high heat. Work in batches to sear the pork shoulder chunks on all sides until browned, adding additional oil as needed. Transfer the seared pork to a plate and set aside.
Heat the remaining tablespoon of oil in the same pan and sauté the onion for 5-7 minutes until softened and translucent. Add the remaining 4 cloves of minced garlic during the last 30 seconds until fragrant.
Return the pork to the Dutch oven with the sauteed onion and add the blended chili sauce and bay leaf. Mix everything together until it is evenly distributed.
Bring to a boil, then reduce heat to low and simmer, covered, for 2 to 2½ hours until the pork is fork tender and done. This can also be done in a 325°F oven.
Serve in a bowl with white rice, warm tortillas, and/or pinto beans. Garnish with chopped cilantro and lime wedges.