Heat your neutral oil in the pressure cooker pot over medium heat. Toss in chuck and oxtail, searing them on all sides until got that nice brown crust.
Take the meat out and set aside so it can rest a sec. In the same pot, toss in the ancho, guajillo, and chiles de arbol. Toast those for about 2 minutes, stirring to get all that smoky smell going.
Add the halved onion, garlic cloves, and roma tomatoes to the pot, stirring around for another 2 minutes to bring out their flavors.
Put the meat back in the pot and pour in enough water or beef broth to cover everything just right.
Lock the lid with the sealing ring in place and set the valve to sealing. Let it pressure build on high, then cook for about 50 minutes.
After cooking, use natural release for 15 minutes to keep that moisture locked in, then do a quick release to let out the rest of the steam.
Remove the meat and shred it into bite-sized pieces with forks. Strain the broth if you want it extra smooth and return it back to the pot. Toss the shredded meat back in and simmer a few minutes to get it soaked in all that broth flavor.
Warm up your tortillas, sprinkle on the Oaxacan cheese, fill them up with the birria, and serve with the broth on the side for dipping. Squeeze fresh lime over top and sprinkle some minced onion and cilantro for that perfect finish.
Notes
Store leftover birria in an airtight container in the fridge. It stays tender and juicy for up to 4 days, perfect for quick dinners or lunch tacos.