Prep chiles: Remove stems and seeds from all dried chiles. Toast them in a dry skillet until fragrant. Soak in hot water for 15 minutes.
Make the sauce: Using the saute function on the Instant Pot, add a splash of oil, then add tomatoes, onions, garlic, toasted/soaked chiles, bay leaves, cinnamon stick, thyme, oregano, cumin, smoked paprika, peppercorns, and chipotles in adobo. Sauté until softened and aromatic (about 5–7 minutes). Add 2 cups water, simmer for 7 minutes more, then discard bay leaves and cinnamon stick.
Blend: Carefully transfer the mixture to a blender, blend until smooth. (Add a bit more warm water if needed for blending.)
Combine: Return sauce to Instant Pot. Add both short rib and chuck roast, salt, beef bouillon, and enough water to just cover the meat. Stir well.
Pressure Cook: Seal the Instant Pot. Cook on high pressure for 45 minutes, natural release for 20 minutes. (You want the meat very tender.)
Shred the meat: Remove meat to a bowl and shred, discarding bones and extra fat. Skim as much fat off the top of the broth as you can (reserve for frying tacos).
Make the tacos: Dip tortillas quickly into the braising liquid, fry in a skillet with a little reserved fat. Add shredded meat, plenty of cheese, and cook until crisp. Fold into tacos.
Serve: Garnish tacos with chopped cilantro, diced onion, and serve with a bowl of the birria consomé for dipping. Squeeze lime over everything just before eating!