Pat the short ribs dry, if needed, and sprinkle them all over with salt and pepper.
Finely dice the onion, celery, and carrots.
Warm half of the oil in a skillet over medium-high heat and sear the short ribs on all sides in two batches so you don't overcrowd the skillet. Make sure to sear until you have a nice mahogany crust.
Put the short ribs in the slow cooker once browned, in one layer with the bones facing up.
Add the onion, celery, and carrots to the skillet along with the remaining oil. Reduce the heat slightly, as needed, so the vegetables don't burn.
Cook the vegetables for a couple minutes then add the garlic, cook a minute more then add the wine and deglaze the pan, scraping any browning from the skillet.
As it starts to simmer, add the balsamic vinegar, tomato paste, broth, rosemary, and bay leaf. Bring to a simmer then pour over the short ribs.
Turn the slow cooker on low and cook for 6 - 8 hours.
Remove the bay leaves when it is done and discard it.
Serve over mashed potatoes or thick pasta like pappardelle.