Preheat oven to 300 degrees.
Set a dutch oven or oven-proof lidded pot over medium-high heat. Swirl with olive oil. Pat the short ribs dry with a paper towel and season meat well on all sides with salt and pepper. Working in batches, sear the short ribs on all sides until nicely browned. Remove short ribs to a plate and set aside.
Add the onions, carrots, and celery to the pan. Cook, stirring often, about 5 minutes. Add the garlic and cook 1-2 more minutes. Return the ribs to the pan, followed by broth, vinegar, sugar, and tomatoes. Season with salt and pepper. Bring to a simmer.
Cover and put into the oven. Bake for 3 hours.
Serve short ribs over mashed potatoes with some of the vegetables and pan juices.