Pat the short ribs dry with paper towels and season generously with salt and pepper.
Heat olive oil in a large Dutch oven or oven-safe pot over medium-high heat. Sear the short ribs on all sides until deeply browned. Remove the ribs from the pot and set aside.
Add the chopped onion, carrots, and celery to the pot. Cook, stirring occasionally, until softened, about 5-7 minutes.
Add the minced garlic and tomato paste to the pot and cook for 1 minute more, until fragrant.
Pour in the red wine, scraping up any browned bits from the bottom of the pot. Let it simmer and reduce by about half, about 5 minutes.
Return the short ribs to the pot. Add the beef broth, thyme sprigs, and bay leaf. The liquid should come about two-thirds of the way up the ribs. If not, add more broth or water.
Bring the liquid to a simmer, then cover the pot tightly and transfer to the preheated oven.
Braise for 2.5 to 3 hours, or until the short ribs are fork-tender.
Remove the short ribs from the pot and set aside. Skim off any excess fat from the surface of the sauce. Discard the thyme sprigs and bay leaf.
If desired, thicken the sauce by simmering it on the stovetop over medium heat until it reaches your desired consistency. Season with additional salt and pepper to taste.
Return the short ribs to the sauce to warm through before serving.
Notes
Serve with mashed potatoes, polenta, or crusty bread to soak up the delicious sauce.