Preheat a 6- or 8-quart Instant Pot® on the saute setting until the display reads "hot" (see Cook's Note). Sprinkle the chuck roast and pork shoulder all over with 1 teaspoon of salt and a generous amount of pepper. Add 2 tablespoons oil to the pot, then add the chuck roast and brown on all sides, about 2 minutes per side. Remove to a sheet pan or large plate.
Add more oil to the pot, if necessary, and repeat the process with the pork shoulder, browning on all sides. Remove and set aside with the beef.
Add the onion and poblano to the pot along with 1 teaspoon salt and cook, stirring occasionally, with a wooden spoon and scraping the browned meaty bits from the bottom of the pot, until the onion is translucent, about 5 minutes. Add the chili powder, cumin, paprika, oregano and cook, stirring to evenly distribute and heat the spices until fragrant, about 1 minute. Add the tomato paste and garlic and continue to cook, stirring, 2 minutes more. Add the tomatoes and beef broth and stir to combine, scraping the bottom with your spoon. Turn the pot off.
Cut the chuck and pork shoulder into 3/4-inch pieces. Add to the pot along with the chorizo and stir to combine. Follow the manufacturer's guide for locking the lid and preparing to cook. Set to pressure cook on high for 30 minutes. After the pressure cook cycle is complete, follow the manufacturer's guide for quick release and wait until the quick release cycle is complete. Be careful of any remaining steam and unlock and remove the lid. Stir the chili and serve with toppings such as sour cream, shredded Cheddar and chopped cilantro.
Settings may vary on your Instant Pot® depending on the model. Please refer to manufacturer's guide.
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