Season the chicken breasts with salt and black pepper on both sides.
Heat the olive oil in a cast iron skillet over medium-high heat. Add the chicken breasts and cook for 6-7 minutes on each side, or until golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the sliced mushrooms and cook for about 5 minutes, or until they are browned and tender. Stir in the minced garlic and cook for an additional 1 minute.
Pour in the chicken broth, scraping the bottom of the skillet to release any browned bits. Bring to a simmer and let it cook for 2-3 minutes.
Reduce the heat to low and stir in the heavy cream, dried thyme, and dried rosemary. Allow the sauce to simmer for another 3-4 minutes until it thickens slightly.
Stir in the grated Parmesan cheese until melted and well combined. Return the cooked chicken to the skillet, coating it in the creamy sauce. Cook for an additional 2 minutes to heat through.
Garnish with fresh parsley before serving.