Season the chicken pieces with salt and pepper. Heat olive oil in a large, deep skillet over medium-high heat. Add chicken and cook until browned on all sides, about 5-7 minutes. Remove chicken from the skillet and set aside.
Add mushrooms to the same skillet and cook until they release their liquid and start to brown, about 5 minutes. Add the chopped onion and cook until softened, about 3 minutes.
Stir in the minced garlic and dried thyme; cook for 1 minute until fragrant.
Add the uncooked rice to the skillet and stir for 1 minute to toast it lightly.
Pour in the chicken broth. Bring the mixture to a simmer, scraping up any browned bits from the bottom of the pan.
Reduce heat to low, cover the skillet, and cook for 15 minutes, or until the rice is tender and most of the liquid is absorbed. Do not lift the lid during this time.
Remove the skillet from the heat. Stir in the heavy cream and Parmesan cheese until the sauce is smooth and creamy.
Return the cooked chicken to the skillet and stir gently to coat. Cover and let stand for 5 minutes to allow the chicken to heat through and the sauce to thicken slightly.
Garnish with fresh parsley before serving.
Notes
For the creamiest sauce, stir the cream and cheese in off the heat. This prevents the dairy from separating or curdling.
Use low-sodium broth if you prefer to control the salt content, as Parmesan cheese adds saltiness.
If the rice seems too thick before the 15 minutes are up, add a splash more broth or water.