In a large container, submerge ribs in brine and refrigerate for at least 12 and up to 24 hours. When ready to cook, remove from brine and pat dry.
Heat oven to 325°F. Place an ovenproof stockpot or Dutch oven over medium-high heat and coat with oil. Add short ribs in batches, brown well (4–5 minutes per side), and remove to a plate. Reduce heat to medium, add onion, carrots, leek and garlic to pan and cook until onion is tender, about 5 minutes. Add flour and cook for 3 minutes, stirring often. Add the thyme and wine, scraping up browned bits, then raise heat to high and let the wine boil until it reduces to a thick, almost syrupy, consistency. Add the short ribs and beef broth to the pan, return to a boil, then cover tightly and place in the oven until ribs are very tender, about 3 hours.
Carefully remove the short ribs from the pan (the meat will be falling off the bone) and set aside. Strain the braising liquid through a fine-mesh strainer, pressing on the solids to fully extract the sauce. (Ribs and sauce can be prepared up to this stage and refrigerated in separate containers, for up to two days.)
Return sauce to pan and cook over medium-high heat until it reduces to a gravy consistency. Season with salt, if needed. Reduce heat to low, add kale and cook until tender, 8-10 minutes. Return ribs to pot and warm through before serving. Add vinegar and parsley just before serving.