Preheat oven to 350°F. Season short ribs with salt and pepper. Heat oil in a large Dutch oven over medium-high heat. Brown short ribs on all sides, about 5 minutes.
As the Ribs are browning add the garlic, raw side down. Brown slightly and let the garlic flavor the oil. Add the thyme and tomato paste. When the ribs are fully browned, add the stock and wine. And the full sprig of rosemary. Let the wine cook out a bit before covering the pan and placing in an oven.
Cook until short ribs are tender, 2-2 1/2 hours. Transfer short ribs to a platter. Strain sauce from pot into a pan over medium heat. Smash the garlic and herbs through the strainer to further flavor the Sauce. Bring the sauce to a simmer. Spoon fat from surface of sauce and discard; season sauce to taste with salt and pepper. Spoon the sauce over the ribs, mushrooms and polenta.