Preheat the oven to 375°F (190°C). Season the chicken breasts with salt and pepper, then dredge them in flour, shaking off any excess.
In a large skillet, heat the olive oil over medium-high heat. Add the chicken breasts and cook for about 5 minutes on each side, or until golden brown. Remove the chicken from the skillet and set aside.
Pour in the chicken broth and bring to a simmer. Stir in the heavy cream and thyme, then return the chicken to the skillet.
Sprinkle the mozzarella and Parmesan cheese over the chicken. Transfer the skillet to the preheated oven and bake for 20-25 minutes, or until the chicken is cooked through and the cheese is bubbly and golden.
Remove from the oven and let it rest for a few minutes. Garnish with fresh parsley before serving.
Notes
For a lighter version, substitute the heavy cream with half-and-half or a non-dairy alternative.
Add spinach or sun-dried tomatoes to the mushroom mixture for extra flavor and nutrition.