Pat the short ribs dry with paper towels, and generously season them with salt and pepper. Set aside.
Set a large (6-quart is good) electric Instant Pot on saute setting. Once the setting has reached hot, sear as many short ribs as possible without overcrowding the pan. Sear them until golden brown (about 3 minutes per side). Repeat with the rest of the ribs, adding olive oil as needed. Transfer your short ribs to a plate and set aside.
Add the onion, garlic, carrots, and celery to the Instant Pot. Saute for 2-3 minutes or until the vegetables soften. Next, add tomato paste and brown sugar. Cook, stirring frequently, until thoroughly combined.
Pour the wine and broth in to deglaze the pot. Stir and scrape the pan's bottom to get all those delicious browned bits. Simmer for 1-2 minutes, then add the thyme and rosemary.
Return the beef ribs into the Instant Pot in an even layer and pour in the juices that accumulated on the plate. Secure the lid and make sure the pressure valve is sealed. Set to cook on Manual/High Pressure for 45-50 minutes.
When the cooking time has finished, set the pressure cooker to Natural Release and leave it undisturbed for 10-12 minutes. After that, turn the steam release knob to the venting position to release the remaining pressure. When the pin in the lid drops (indicating all pressure has been released), open the pot and carefully remove the ribs with tongs to a plate and cover to keep warm.
Carefully skim the excess fat from the top of the cooking liquid using a spoon. Remove the thyme and rosemary and discard. Then, stir the balsamic vinegar into the sauce.
Turn the Instant Pot knob to saute and bring the sauce to a simmer. Make the cornstarch slurry in a small mixing bowl with about 2 tablespoons of the beef broth and the cornstarch. Whisk to thoroughly combine and form the slurry. Slowly pour the cornstarch slurry into the liquid and whisk as the sauce thickens. Continue adding the cornstarch until the sauce reaches the desired thickness.
Season with salt and pepper to taste.
Return the short ribs to the pot, season with green onions, and serve them over your favorite creamy mashed potatoes and a spoonful or two of the gravy.