Heat the oil in a heavy 5 1/2-quart saucepan over medium heat. Add the celery, carrot, and onion. Saute the vegetables until the onion is translucent, about 5 minutes. Season with salt and pepper, to taste. Stir in the tomatoes with their juices, chicken broth, tomato paste, bay leaf, Italian seasoning and thyme. Add the chicken breasts; press to submerge.
Bring the cooking liquid to a simmer. Reduce the heat to medium-low and simmer gently uncovered until the chicken is almost cooked through, turning the chicken breasts over and stirring the mixture occasionally, about 25 minutes. Using tongs, transfer the chicken breasts to a work surface and cool for 5 minutes. Discard the bay leaf. Add the kidney beans to the pot and simmer until the liquid has reduced into a stew consistency, about 10 minutes.
Discard the skin and bones from the chicken breasts. Shred or cut the chicken into bite-size pieces. Return the chicken meat to the stew. Bring the stew just to a simmer. Season with salt and pepper, to taste.
Ladle the stew into serving bowls, top with Parmesan cheese and serve with the bread.