Preheat oven to 350°. Season short ribs with salt and pepper. Heat oil in a large Dutch oven over medium-high. Working in 2 batches, brown short ribs on all sides, about 8 minutes per batch. Transfer short ribs to a plate. Pour off all but 3 Tbsp. drippings from pot.

Add onions, carrots, and celery to pot and cook over medium-high heat, stirring often, until onions are browned, about 5 minutes.

Add flour and tomato paste; cook, stirring constantly, until well combined and deep red, 2–3 minutes. Stir in wine, then add short ribs with any accumulated juices.

Bring to a boil; lower heat to medium and simmer until wine is reduced by half, about 25 minutes. Add all herbs to pot along with garlic. Stir in stock. Bring to a boil, cover, and transfer to oven.

Cook until short ribs are tender, 2–2½ hours. Transfer short ribs to a platter. Strain sauce from pot into a measuring cup. Spoon fat from surface of sauce and discard; season sauce to taste with salt and pepper. Serve in shallow bowls over mashed potatoes with sauce spooned over.