Gather ingredients. For a recipe like birria, it helps to gather your ingredients before getting started. It will keep you feeling organized.
Prepare the chilies. Use kitchen scissors to cut open each chili and remove the seeds. Then, heat a large skillet over medium heat and toast the chilis until fragrant (1-2 minutes or so) taking care not to toast too long as dried chilies will quickly turn bitter. Transfer the chilies to a medium pot filled with simmering water and soak until rehydrated, approximately 15-30 minutes.
Brown the onion and tomatoes (see notes). Add one tablespoon of olive oil to the same skillet over high heat. Add the onions and let them cook, undisturbed, for 1-2 minutes. Mix well and continue to cook for an additional minute or so. Add the crushed garlic, sauté for 30-60 seconds, then remove everything to a clean plate. Add the whole tomatoes, cook on each side for 2 minutes and remove to the same plate as the onions.
Toast the whole spices. Add the remaining tablespoon of olive oil to your pan. Add the cumin seeds, black peppercorns, whole cloves, and cinnamon stick. Cook, mixing continuously, for 1-2 minutes, or until fragrant. Return the onions, garlic and tomatoes to the pan with the toasted spices and mix well to combine. Remove from heat.
Add the dried herbs and seasoning. Return the onions, garlic, and tomatoes to the toasted spices and mix well to combine. Stir in the ginger, thyme, oregano, and marjoram. Remove from heat.
Blend. Transfer the onion, tomato, and spice mixture to a large, high-speed blender (I have a Vitamix so even the cinnamon stick went in there). Transfer the rehydrated chilies plus 1-2 cups of the soaking liquid to the blender and process until smooth.
Cook. Transfer the prepared chili sauce to a large, deep Dutch oven set over medium-high heat. Add the bay leaves and stir in the orange just and apple cider vinegar. Add the beef and lamb and bring to a simmer. Cover, reduce heat to medium-low, and cook for at least 3-4 hours.
Serve. When ready to serve, transfer to a large bowl and serve garnished with chopped white onion, cilantro, and a squeeze of fresh lime juice.