Cut the chicken breasts horizontally into thin cutlets and pat them completely dry with paper towels. In a small bowl, combine the salt, pepper, garlic powder, onion powder, and dried thyme. Toss the chicken cutlets with olive oil and the seasoning mixture until evenly coated.

Heat a large skillet over medium-high heat. Add the seasoned chicken cutlets in a single layer, working in batches if necessary. Sear for 4 to 5 minutes on the first side without moving them. Flip and cook for another 4 to 5 minutes until golden and the internal temperature reaches 165°F. Remove to a plate and set aside.

Add the butter and olive oil to the same skillet over medium-high heat. Once the butter is melted and foaming, add the sliced mushrooms in an even layer. Let them cook undisturbed for 3 to 4 minutes, then stir and continue cooking for another 4 to 6 minutes until deeply golden brown and any liquid has evaporated.

Add the diced onion to the mushrooms and sauté for about 3 minutes until translucent and soft. Stir in the minced garlic and cook for 1 minute, stirring constantly until fragrant.

If using white wine, pour it into the skillet and use a wooden spoon to scrape up all the browned bits. Let the wine simmer for about 2 minutes until reduced by half. Add the chicken broth and bring to a simmer. Cook for about 5 minutes until the liquid is reduced by roughly one-third.

Lower the heat to medium-low so the sauce is gently simmering. Stir in the heavy cream and grated Parmesan cheese, whisking until the cheese melts completely and the sauce becomes smooth and velvety. Taste and adjust seasoning with additional salt and pepper if needed.

Return the chicken cutlets and any accumulated juices from the plate back to the skillet, nestling them into the creamy mushroom sauce. Let everything simmer together for 2 to 3 minutes so the chicken warms through. Garnish with chopped fresh parsley and serve immediately.