Set Instant Pot to sauté setting and heat oil until hot. Pat short ribs dry and generously season with kosher salt and freshly cracked pepper. Add to heated insert and brown well, without moving except to flip, about 6 minutes a side. Remove to plate and set aside.
Add carrots, shallots and a sprinkle of salt to remaining oil in pot. Cook until starting to soften and brown slightly. Add garlic, stir until just fragrant. Turn off sauté mode.
Add red wine to pot and stir to scrape up any browned bits. Add the chicken stock and stir to combine. Season with a little more salt and pepper. Nestle the short ribs and any accumulated juices back into the pot. Top with thyme sprigs, and rosemary, if using.
Place lid on pot and seal. Cook on regular pressure cook mode set at high pressure for 45 minutes. When done, let pressure naturally release for 15 minutes, then manually release any remaining pressure.
Carefully remove ribs to plate and tent with foil. If needed, skim any excess fat off the liquid and discard. Turn on sauté mode, add balsamic vinegar and boil for about 5 to 10 minutes until reduced slightly and the sauce starts to turn syrupy. Turn off sauté mode. Spoon sauce over short ribs and serve.