Preheat the oven to 325°. Heat 2 tablespoons of the olive oil in a large enameled cast-iron casserole. Season the ribs with salt and pepper. Add half of the short ribs to the casserole and cook over moderately high heat until browned, about 6 minutes. Transfer the ribs to a plate. Repeat with the remaining oil and ribs.
Add the celery, carrot and onion to the casserole and cook over moderate heat until softened, about 7 minutes. Add the tomato paste and cook, stirring, until glossy, about 2 minutes. Add the thyme sprigs, anchovies, bay leaf and garlic and cook, stirring, for 2 minutes. Add the stock, wine and vinegar and bring to a boil. Return the short ribs to the casserole, then cover them and braise in the oven until the meat is very tender, about 3 hours.
Transfer the ribs to a platter. Strain the braising liquid, pressing hard on the solids; skim the fat. Season with salt and pepper. Serve half of the ribs with some of the sauce. Reserve the rest in the remaining sauce for sandwiches.
Make Ahead The short ribs can be refrigerated in their sauce for 3 days. 1/