01 - Set the oven to 325°F to prepare for braising.
02 - Pat the short ribs dry and season generously with salt and pepper. Lightly coat with flour, shaking off any excess.
03 - Heat olive oil in a large Dutch oven over medium-high heat. Sear ribs on all sides until deeply browned, about 2 to 3 minutes per side. Remove and set aside.
04 - Add diced onion, carrots, and celery to the pot; cook until softened, approximately 5 minutes. Stir in minced garlic and tomato paste, cooking for 1 minute more.
05 - Pour in red wine, scraping the bottom to release browned bits. Simmer gently for 5 minutes to reduce slightly.
06 - Incorporate beef stock, thyme, rosemary, and bay leaves. Return short ribs along with any accumulated juices to the pot and bring to a gentle simmer.
07 - Cover the pot and transfer to the oven. Cook for 2 to 2.5 hours until ribs are tender and falling off the bone.
08 - About 30 minutes before ribs finish, place potatoes in salted boiling water and cook until fork-tender, 15 to 20 minutes. Drain, return to pot, and mash with butter and milk. Season with salt and pepper.
09 - Remove ribs from the oven. Discard herb stems and bay leaves. Skim off excess fat from sauce. Serve ribs atop mashed potatoes, spooning sauce and vegetables over the top.