Heat the oven to 350 °F. Set the rack so the covered Dutch oven sits level and has room for heat to circulate.
Warm 1 tablespoon of olive oil in a large Dutch oven or oven-safe pot with a lid over medium-high heat, then cook the onion and carrots for a few minutes while stirring. When the vegetables smell sweet and look lightly softened, transfer them to a plate.
Season the short ribs with salt and pepper, then coat every side with flour. The flour should cling in a thin, even layer with no thick patches that could burn.
Add the remaining 2 tablespoons of olive oil and raise the heat to high, then brown the ribs for about 45 seconds per side. Work in batches if needed so the beef forms a dark crust instead of releasing liquid and steaming.
Move the browned ribs to a plate and pour the wine into the hot pot. Scrape the browned fond from the bottom as the wine bubbles, then let it boil for 2 minutes.
Pour in the beef broth, season with salt and freshly ground black pepper, and stir in the tomato paste. Taste the liquid before adding the ribs back, adjusting the seasoning if it needs more savory depth.
Return the ribs and vegetables to the pot, then add the whole thyme and rosemary sprigs. The ribs should sit in the braising liquid with their tops still visible.
Cover the pot and braise at 350 °F for 2 hours, then lower the oven to 325 °F and continue cooking for 30 to 45 minutes. The meat is ready when it is nearly falling from the bone and a paring knife slides into the center with almost no resistance.
Keep the lid on and let the pot rest for 20 minutes before serving, then skim the fat that rises to the surface. As the steam settles, the separated fat becomes easier to spot and remove with a spoon.
Spoon the ribs over mashed potatoes and finish each portion with gravy from the pot. The meat should pull apart easily while the potatoes catch every bit of the savory sauce.
Transfer leftover short ribs to a sealed container and refrigerate for 3-4 days depending on the freshness of your ingredients. Keep the ribs nestled with some braising liquid so they stay moist when warmed.