Preheat oven to 325°. Rub short ribs all over with salt and pepper. Heat a 6- to 8-qt. dutch oven over medium-high heat. Add oil and brown short ribs in 2 batches on all sides, turning as needed, 10 minutes per batch. Transfer short ribs to a large plate and set aside.
Reduce heat to medium and add onions, carrots, celery, and garlic to pot. Cook, stirring often, until vegetables are soft and lightly browned, about 25 minutes. Add tomato paste, wine, broth, thyme, and bay leaves and stir to release any brown bits. Bring to a boil and cook about 5 minutes to concentrate flavors. Return short ribs to pot and cover.
Braise in oven until ribs are nearly tender when pierced with a fork, 1 1/2 to 2 hours. Uncover and cook until a fork slides right into meat, 30 minutes more. With a slotted spoon, transfer ribs to a 9- by 13-in. baking dish. Set ribs aside.
Strain juices from pot into a bowl, pressing vegetables to release liquid. Discard vegetables and return juices to pot.
Increase oven to 400° and set rack in upper third. Bring juices to a boil over high heat, tip pot, and skim fat with a shallow spoon. Boil until sauce is reduced to about 2 cups, 20 to 25 minutes.
Ladle about 1 cup sauce over ribs (save remaining sauce for other uses). Roast ribs, spooning sauce over them several times, until heated through and glazed, 15 to 25 minutes. Set meat on plates, spoon on sauce, and top with a little watercress.
Make ahead: Through step 4, up to 1 day, chilled, with meat added to juices in pot. In step 5, lift fat from cold juices; then transfer meat to baking dish and reduce juices (no need to skim fat).