Peel onion and cut lengthwise into 8 wedges. Rinse celery and tomatoes. Trim and discard ends of celery and cut into 3-inch lengths (reserve leaves for garnish); core tomatoes and cut in half lengthwise.
Rinse chicken and pat dry. Sprinkle lightly all over with salt and pepper.
Pour oil into a 10- to 12-inch nonstick frying pan over high heat, tilting to coat bottom. When oil is hot, add chicken and turn as needed to brown lightly on both sides, 4 to 5 minutes total. Transfer to a rimmed plate.
Add onion, celery, and tomatoes to pan; turn vegetables occasionally until lightly browned, 3 to 5 minutes total. Return chicken and any accumulated juices to pan and add broth and thyme. Cover, reduce heat to low, and simmer until chicken is no longer pink in center of thickest part (cut to test), 6 to 8 minutes.
Transfer breast halves to wide, shallow bowls; ladle vegetables and broth around chicken. Add more salt and pepper to taste. Garnish with celery leaves.