Preheat oven to 325 degrees F.
Pat ribs dry with paper towels and season with salt and pepper on all sides.
Heat 2 tablespoons of butter in a large Dutch oven over medium-high heat. Add the ribs and brown on all sides. Transfer the ribs to a plate, then add the garlic and stir until softened but not brown, about 1 minute. Carefully pour in the wine, bring to a boil and cook until reduced to about 1 cup. Add the ribs and any juices they have released back to the pot along with the stock. Reduce the heat and bring to a simmer, then cover the pot and transfer it to the oven, to braise, until the meat easily falls off the bone, about 3 hours.
Transfer the ribs to a plate and tent with foil to keep warm. Strain the braising liquid through a fine mesh sieve into a saucepan set over medium-high heat. Simmer the liquid, skimming fat as necessary, until thickened, about 10 minutes. Remove from the heat and whisk in the remaining 1 tablespoon of butter. Taste for seasoning and adjust, if necessary. Transfer the ribs and sauce to a serving dish and serve.