Place 6 dried Guajillo chiles and 4 dried Ancho chiles in a bowl filled with near-boiling water. Let them soak for at least 30 minutes until softened.
While the chiles soak, season 3 lbs of chuck roast (cut into 4 chunks) with kosher salt and ground black pepper to taste. Heat 1 tbsp of vegetable oil in a large Dutch oven over medium-high heat. In batches, sear the chuck roast on all sides, adding another tbsp of oil as needed. Transfer the seared meat to a plate and set aside.
Add the remaining 1 tbsp of oil to the Dutch oven. Toss in 3 quartered vine tomatoes, 1 quartered white onion, and 8 peeled and smashed garlic cloves. Sauté for 2-3 minutes until fragrant.
Pour in 4 cups of low-sodium beef broth, scraping up any browned bits from the bottom of the pot.
Drain the water and transfer the softened chiles from the water to the pot. Add 1 1/2 lbs of bone-in short ribs, along with 1 tbsp of ground coriander, 2 tsp of ground cumin, 2 tsp of dried oregano, 1 tsp of chili powder, and 1 cinnamon stick. Bring the mixture to a boil, then reduce the heat to maintain a gentle simmer for 30 minutes.
Remove the short ribs and set them aside with the seared roast. Skim any impurities from the top of the pot and discard the cinnamon stick. Transfer 1 1/2 to 2 cups of the braising liquid to a bowl, cover, and refrigerate. Use an immersion blender to blend the remaining broth until smooth.
Return the seared chuck roast, short ribs, and any juices to the Dutch oven. Stir in 1/2 cup of orange juice and 1/4 cup of apple cider vinegar. Season with additional salt and pepper to taste. Bring the mixture to a simmer, cover, reduce heat to medium-low, and cook for 3-4 hours until the meat is tender.
Once fully cooked, remove the bones from the short ribs and discard them. Use two forks to shred all the meat into the sauce.
Remove the reserved braising liquid from the refrigerator. Working one at a time, dip each of the 15 corn tortillas into the liquid and place them in a large skillet set over medium heat. Sprinkle 1-2 tbsp of shredded Monterey Jack cheese onto one half of each tortilla. Top with 2-3 tbsp of shredded beef, a sprinkle of 1 cup of chopped yellow onion, and 1/2 cup of chopped fresh cilantro.
Fold the tortilla in half. Cook each taco for 30-45 seconds per side, until the exterior is crispy and golden brown. Transfer to a serving platter.
Squeeze fresh lime wedges over the tacos and serve with any remaining braising liquid on the side for dipping. Enjoy your homemade birria tacos!