Season chicken breasts with salt and pepper on both sides. Optionally, coat lightly with all-purpose flour for extra texture.
Heat olive oil or butter in a large skillet over medium heat. Cook chicken breasts for about 5-6 minutes per side, until golden brown and fully cooked through. Remove and set aside.
In the same skillet, sauté finely chopped onion and minced garlic until soft and fragrant.
Add sliced mushrooms, cooking for 4-5 minutes until tender and slightly caramelized.
Pour in chicken broth, scraping up browned bits from the pan.
Stir in heavy cream and add dried thyme or Italian seasoning. Simmer gently until sauce thickens.
Add grated Parmesan cheese, stirring until melted (optional).
Return chicken breasts to the skillet, spooning sauce over them. Simmer for 3-5 more minutes to heat through.
Garnish with fresh chopped parsley and serve hot.