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Step 1
Dice 1 medium yellow onion (about 1 1/2 cups). Finely chop 2 garlic cloves. Peel and cut 2 medium carrots crosswise into 1-inch pieces.
Step 2
Heat 1 tablespoon vegetable oil in a 6-quart or larger electric pressure cooker on the highest Sauté setting. Meanwhile, pat 4 to 4 1/2 pounds bone-in short ribs dry with paper towels. Trim off any large chunks of surface fat. Season all over with 1 tablespoon of the kosher salt and 1/2 teaspoon black pepper.
Step 3
Add half the short ribs bone-side up to the pressure cooker and sear the meaty side until golden-brown, about 5 minutes. Transfer to a plate. Repeat searing the remaining short ribs and transfer to the plate.
Step 4
Add the onion, garlic, and remaining 1/2 teaspoon kosher to the pressure cooker and cook, scraping up any browned bits from the bottom with a wooden spoon, until the onion is softened, about 5 minutes. Add 2 tablespoons tomato paste, 4 fresh thyme sprigs or 1 teaspoon dried thyme, and 1 bay leaf. Cook, stirring frequently, until the tomato paste is darkened in color, about 1 minute. Add 1/2 cup dry red wine and bring to a simmer, scraping up any browned bits from the bottom of the pot. Turn off the pressure cooker.
Step 5
Add 1/2 cup beef broth and the carrots. Return the short ribs to the pot in an even layer; pour in any accumulated juices on the plate. Lock the lid on and check that the pressure valve is set to seal. Set to cook for 45 minutes on HIGH pressure. It will take 9 to 12 minutes to come up to pressure.
Step 6
When the cook time is up, let the pressure naturally release for 10 minutes. Meanwhile, whisk 2 tablespoons cornstarch and 2 tablespoons water together in a small bowl until the cornstarch is suspended. Coarsely chop some fresh parsley leaves for garnish if desired.
Step 7
Quick release any remaining pressure. Carefully open the pressure cooker and use tongs to transfer the short ribs to a serving platter; tent loosely with aluminum foil to keep warm. Pour the sauce through a fine-mesh strainer into a heatproof bowl or fat separator; discard the contents of the strainer. Let the sauce sit for a few minutes for the fat to rise to the top, then pour or skim the fat off with a spoon. Pour the sauce back into the pressure cooker (wipe it out first if needed) and bring back to a simmer on the Sauté function. (Alternatively, strain into a saucepan, skim off the fat, and bring back to a simmer over medium heat on the stovetop.)
Step 8
Stir the cornstarch slurry again, then add to the sauce and simmer until thickened, about 1 minute. Taste and season the sauce with more kosher salt and black pepper as needed. Rewarm the short ribs in the sauce if desired.
Step 9
Serve the short ribs and sauce over mashed potatoes, polenta, or buttered egg noodles. Garnish with the parsley if desired.