Preheat oven to 450 degrees.
Rub the chicken breasts on all sides with extra virgin olive oil, salt and pepper.
Place them, skin side up, in a roasting pan on the center rack and roast for about 30-40 minutes, or until the thickest part of the breast registers 157 degrees and skin is crispy.
Meanwhile, combine tomatoes and 2 tablespoons oil in a large bowl. Season with salt and pepper; toss to coat. Heat 1 tablespoon oil in a large heavy ovenproof skillet until it shimmers. Carefully add tomatoes to pan. Transfer skillet to oven and roast, turning once, until tomatoes burst and give up some of their juices, about 15 minutes. Place tomatoes back in the bowl and drizzle with Worcestershire sauce.
Add remaining tablespoon of oil to same skillet; heat over medium heat. Add shallot and cook, stirring often, until fragrant, about 1 minute. Deglaze pan with vinegar; add tomatoes and their juices and simmer until sauce is just beginning to thicken, about 1 minute. Season with salt and pepper.
When chicken is done, transfer to a cutting board and let rest for at least 5 minutes. Cut along the bone to remove meat. Pull off any remaining pieces, but don’t throw away bones. You can use them as a base for chicken stock. Slice chicken and divide between two plates. Spoon tomatoes and sauce over and serve.