01 - Pat chicken breasts dry and season both sides with salt and black pepper.
02 - Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Sear chicken breasts for 3 to 4 minutes per side until golden and just cooked through, then transfer to a plate and loosely cover with foil.
03 - Add remaining olive oil and butter to the skillet. Sauté onions and mushrooms for 5 to 6 minutes until softened and golden, then add minced garlic and cook for 1 minute.
04 - Sprinkle flour over the mushroom mixture and stir continuously for 1 minute to eliminate raw flour taste.
05 - Gradually pour in chicken broth, scraping up any browned bits from the skillet. Add dried thyme, bring to a gentle simmer, and cook for 2 minutes.
06 - Stir in heavy cream and Parmesan cheese. Simmer for 2 to 3 minutes until the sauce thickens slightly.
07 - Return chicken breasts to the skillet, spoon sauce over them, and simmer on low heat for 3 to 5 minutes until thoroughly heated.
08 - Sprinkle with fresh parsley and serve immediately.