If your short ribs have any large, thick pieces of fat attached to them, trim those off. Don’t trim all the fat though, we want some for flavor!
Sprinkle the ribs on both sides with the salt.
Heat the olive oil in a large, heavy pot over medium heat.
Arrange the beef ribs, meaty side down, so that they fit without overlapping.
Cook the short ribs for 5 minutes.
Turn each piece and cook them for another 5 minutes.
Remove the ribs from the pot and set them on a plate or pan. Keep them warm by loosely covering with a piece of foil, and/or placing them in the microwave or oven.
In the same pot, still over medium heat, add the onions, carrots and celery.
Cook the vegetables for 5 minutes, stirring frequently.
Add the garlic and oregano and cook them for 1 minute while stirring.
Carefully add the red wine to the vegetables and stir, while gently scraping any bits off the bottom of the pot. Keep at a simmer until most of the red wine has cooked out, approximately 2-3 minutes.
Finish the sauce by adding the tomato sauce and the beef broth. Stir well.
Place the short ribs back into the pot, meaty side down. Raise the heat to medium-high and bring the sauce to a simmer.
Cover the pot, lower the heat to medium low. Cook the short ribs for 2 – 2½ hours or until they're fall-apart tender.
Keep the sauce at a simmer, it may be necessary to knock the temperature down (or raise ijust a bit. Stir the sauce occasionally, and check that the liquid is simmering how we want it. Check throughout the cooking time not just at the beginning.