Pound and season the chicken. Lightly pound the chicken to even thickness (about 1/2 inch) so it cooks evenly. Pat dry. Season both sides with salt and pepper, then dust with a thin coat of flour, shaking off excess.

Brown the chicken. Heat olive oil and 1 tablespoon butter in a large skillet over medium-high. Add chicken and cook 4–5 minutes per side until golden and mostly cooked through. Transfer to a plate and tent loosely with foil.

Sauté the mushrooms. In the same pan, add remaining butter and the mushrooms. Spread them out and let them sear undisturbed for 2–3 minutes, then stir. Cook until browned and their moisture has evaporated, about 6–8 minutes. Season with a pinch of salt.

Build aromatics. Reduce heat to medium. Add chopped shallot and cook 2 minutes until softened. Stir in garlic and thyme and cook 30 seconds until fragrant.

Deglaze. Pour in the wine, scraping up brown bits. Let it simmer 1–2 minutes to reduce by about half. If not using wine, use extra broth plus a teaspoon of lemon juice later for brightness.

Make the sauce. Add chicken broth and simmer 2–3 minutes. Stir in cream and Dijon. Bring to a gentle simmer and cook 3–5 minutes until slightly thickened and glossy.

Finish the chicken. Return chicken and any juices to the pan. Simmer on low 3–5 minutes until the chicken is cooked through and the sauce coats the back of a spoon. Adjust salt and pepper. Add a small squeeze of lemon for balance.

Garnish and serve. Turn off heat and stir in chopped parsley. Serve with mashed potatoes, buttered noodles, rice, or crusty bread to catch the sauce.