Season the chicken on both sides with the Italian seasoning, garlic powder, and a generous sprinkling of salt and pepper.
Heat the oil in a large, deep skillet or Dutch oven over medium heat. When the oil is shimmering, add the chicken. Brown until the chicken releases easily from the pan, 2-4 minutes, then turn and brown the other side. Remove the chicken to a plate.
Add the mushrooms and shallot to the pan and saute until golden, 3-4 minutes. Stir in the garlic, sprinkle everything with a little more salt, and saute an additional minute.
Pour a small amount of the chicken broth into the pan and use a wooden spoon or spatula to deglaze, scraping any browned bits off the bottom of the pan. Add the remaining broth, rice, lemon, and thyme, then add the chicken and all the reserved juices back to the pan. Bring to a simmer, then lower the heat, cover, and cook without stirring for 18-20 minutes or until rice is tender and liquid is mostly absorbed.
Slowly stir in cream. It will soak in as you add it. You can start with 1/4 cup and keep adding as it soaks in more. Sprinkle with parsley, if desired. Serve chicken and rice with a sprinkle of Parmesan cheese.