Set the Instant Pot to sauté. When hot, add the oil.
Season the short ribs with salt and pepper and sear on all sides in the pot, about 2 to 3 minutes per side. Work in batches, if necessary, to avoid overcrowding the pot. (You want to sear, not steam). Remove to a plate.
Add the onions, celery, and carrots to the Instant Pot and sauté until onions start getting translucent, about 5 to 7 minutes. Add a pinch or two of salt, and a couple of grinds of pepper.
Add the garlic and sauté another minute or two, making sure the garlic doesn't scorch.
Add the tomato paste, stirring it into the vegetables and let cook for a minute, stirring frequently.
Deglaze the pot with red wine, scraping up any brown bits from the pan, and let cook for 3 or 4 minutes. (When you can’t smell the alcohol from the wine anymore, it’s good).
Add the beef stock to the pot, along with the herbs. Let cook for 3 or 4 minutes.
Add the short ribs back to the pot. Seal and cook on manual high pressure for 35 minutes. When the cook is finished, allow to natural pressure release for 15 minutes.
While the short ribs are cooking, place the mushrooms on a baking sheet or in an oven save dish, adding a little olive oil and salt and pepper, and roast at 400 degrees for about 15 minutes. Reserve the juice, too!
When the natural pressure release time is done, release the pressure and carefully remove the short ribs to a plate. (They should be falling off the bone, but not shredding).
Put a fine mesh strainer over a bowl and strain the liquid in the pot from the vegetables, leaving the vegetables in the strainer.
Skim fat off the top of the liquid or use a fat separator. (Alternately, you may refrigerate for a couple of hours to overnight and scrape the congealed fat off the top).
Set the Instant Pot to sauté and pour the liquid back into the pot.
Mash the aromatics through the strainer, back into the liquid in the pot. (This helps thicken the sauce).
Cook the liquid on sauté, adding the roasted mushrooms (with any accumulated juices), until the sauce reduces, about 5 to 7 minutes.
Add the aged balsamic vinegar and let cook for another minute or two.
Adjust salt and pepper to taste, if needed. If the sauce needs to thicken more, add a little xanthan gum (for low carb) or a cornstarch slurry.
Serve over mashed cauliflower, pasta, or mashed potatoes. Garnish with fresh parsley.