Pat the chicken dry and season with salt and pepper.
Over medium-high heat, heat a 12 inch skillet. When hot add the olive oil. Add the chicken and brown for about 4 minutes on one side, flip and brown for about 3 more minutes. Transfer the chicken to a plate.
Add the onion, carrot, celery, rice, thyme, and 1/4 tsp salt to the skillet. Saute about 3 minutes. Add the beef broth and using a wooden spoon, scrape up all the browned bits on the bottom of the pan.
Reduce the heat to low, cover and simmer for about 25 minutes. Then add the chicken and any juices in the plate. Nestle the chicken pieces in. Cover and cook again for another 20 minutes. Test the rice, it might need another few minutes. Make sure the heat is very low.
Stir in the grapes and walnuts. Taste and adjust the seasonings. Divide between two plates and sprinkle with the parsley to serve.