Preheat your oven to 325°F (163°C).

In a large Dutch oven or heavy pot, heat the olive oil over medium-high heat. Season the beef short ribs with salt and black pepper. Add the ribs to the pot and sear them on all sides until browned, about 3-4 minutes per side. Remove the ribs and set them aside.

In the same pot, add the diced onion and minced garlic. Sauté for about 5 minutes, or until the onion is translucent and fragrant.

Pour in the balsamic vinegar and scrape the bottom of the pot to release any browned bits. Bring to a simmer.

Stir in the beef broth, brown sugar, tarragon, and the seared short ribs. Bring the mixture to a boil.

Cover the pot with a lid and transfer it to the preheated oven. Braise the short ribs for about 2.5 to 3 hours, or until the meat is tender and easily falls off the bone.

Once the ribs are done, remove them from the pot and set them aside. Strain the braising liquid into a saucepan, discarding the solids. In a separate bowl, mix the butter and flour to form a paste. Bring the strained liquid to a simmer and whisk in the butter-flour mixture until the gravy thickens, about 5 minutes.

Plate the beef short ribs and drizzle them generously with the velvety pan gravy. Garnish with additional fresh tarragon if desired.