If the chicken breasts are thick, pound them to an even thickness of about 1/2 inch. Pat dry and season with 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper, then lightly coat each piece in flour and shake off the excess.

Heat the olive oil and butter in a large skillet over medium-high heat. Cook the chicken for about 4–5 minutes per side until golden and the internal temperature reaches 165°F, working in batches if needed, then transfer it to a plate.

Add the sliced mushrooms and chopped onion to the skillet. Cook for 5–7 minutes, until the mushrooms release their liquid and begin to brown, then add the garlic and cook for 30 seconds.

Pour in the dry white wine and scrape the browned bits from the bottom of the skillet. Simmer for 2–3 minutes so the wine reduces slightly and blends with the mushroom and onion flavors.

Add the chicken broth and heavy cream. Simmer for 3–5 minutes, stirring occasionally, until the sauce is slightly reduced.

Stir in the Parmesan, remaining 1/2 teaspoon salt, remaining 1/4 teaspoon pepper, and optional lemon juice. Return the chicken and any accumulated juices to the pan and simmer gently for 2–3 minutes, just until heated through.

Garnish with the chopped parsley and serve immediately. Spoon plenty of the mushroom cream sauce over each chicken breast with rice, mashed potatoes, egg noodles, or crusty bread.