Season and prep the chicken. Pat the chicken dry, then season both sides with salt and pepper.
Lightly dredge in flour if using, shaking off excess.
Brown the chicken. Heat olive oil in a large skillet over medium-high. Add chicken and sear until golden on both sides, about 4–5 minutes per side. Transfer to a plate.
The chicken will finish cooking in the sauce.
Sauté the mushrooms. Lower heat to medium. Add butter, then mushrooms. Cook, stirring occasionally, until they release moisture and turn golden around the edges, 6–8 minutes.
Season with a pinch of salt.
Add aromatics. Stir in shallot and cook 1–2 minutes until softened. Add garlic and thyme; cook 30 seconds until fragrant.
Deglaze. Pour in the wine (if using), scraping up browned bits. Let it reduce by half, about 2–3 minutes.
Add chicken broth and bring to a gentle simmer.
Make it creamy. Stir in heavy cream and Dijon. Simmer 2 minutes to thicken slightly. If using Parmesan, whisk it in until melted and smooth.
Finish the chicken. Return chicken and any juices to the pan.
Simmer gently 5–7 minutes, turning once, until the chicken is cooked through (internal temp 165°F/74°C) and the sauce is silky.
Brighten and season. Add lemon juice to taste. Adjust salt and pepper. Sprinkle with parsley.
Serve. Spoon over mashed potatoes, buttered noodles, rice, or crusty bread.
Add a side of greens for balance.