Pat chuck roast dry with a paper towel. Stir together flour, salt, and pepper in a small bowl. Sprinkle flour mixture all over the roast, pressing lightly to ensure it sticks to the meat.
Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add oil. Add roast to the hot oil and cook, without turning, until browned, about 3 minutes. Turn roast to brown each side, about 3 minutes per side. Transfer roast to a plate.
Add onion, garlic, and minced rosemary to the pot; cook, stirring often, until onion is translucent, about 6 minutes. Add beef stock, wine, and tomato paste and simmer while scraping the browned bits of food off the bottom of the pan with a wooden spoon. Return roast to the pot along with bay leaf.
Close and lock the lid, select Meat/Stew function, and set timer for 40 minutes.
Release pressure carefully using the quick-release method according to manufacturer’s instructions, about 5 minutes. Unlock and remove the lid. Transfer roast to a plate and tent with aluminum foil for 15 minutes.
Add carrots, pearl onions, and celery to the pot. Close and lock the lid. Select high pressure according to manufacturer’s instructions; set timer for 4 minutes. Allow 5 to 10 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer’s instructions, about 5 minutes. Unlock and remove the lid.
Slice roast against the grain, or use 2 forks to pull meat into chunks. Transfer roast, vegetables, and sauce to a serving platter. Discard bay leaf and garnish with rosemary sprigs.
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