Carefully cut open each dried chili pepper to remove the seeds. Toast the chiles in a large skillet set over medium heat until fragrant, then place them in a pot filled with simmering water for 15-30 minutes or until rehydrated.
In the same skillet, brown the onions in one tablespoon of olive oil for 1-2 minutes. Add the garlic cloves and sauté for 30-60 seconds or until fragrant. Remove everything to a clean plate and set aside.
With the same skillet still set over medium-high heat, add the whole tomatoes and cook on each side for approximately 2 minutes. Remove the tomatoes to the same plate as the onions and garlic.
Add one more tablespoon of olive oil to the skillet. Add the cumin seeds, black peppercorns, whole cloves, and cinnamon stick. Cook for 1-2 minutes, mixing often.
Return the onions, garlic, and tomatoes to the toasted spices, then stir in the ginger, thyme, oregano, and marjoram. Remove from heat.
Transfer the onions, tomatoes, and all the spices to a high-speed blender, plus the rehydrated chilies and 1-2 cups of the soaking liquid. Blend until smooth.
Transfer the blended chili sauce to a large pot or Dutch oven set over medium-high heat. Add the orange juice, apple cider vinegar, beef, and bay leaves. Bring to a gentle simmer, then cover with a lid and cook for at least 3-4 hours.
Remove the beef from the pot and shred it with a fork. Save the birria broth.