Prepare the chicken: Pat chicken dry and season evenly with salt and pepper. If thick, pound to an even thickness.
Sear: Heat 1 tbsp olive oil in a large skillet over medium-high heat. Sear chicken for 5–7 minutes per side, or until golden and the internal temperature reaches 74°C (165°F). Transfer to a plate.
Cook the mushrooms: Add remaining olive oil and butter to the same skillet. Add mushrooms and sauté for 5–6 minutes until browned. Add onion, if using, and cook until softened. Stir in garlic and thyme; cook for 30–60 seconds.
Deglaze: Add chicken stock, scraping up the browned bits from the pan. Simmer for 2–3 minutes. If using wine, add it first and reduce slightly before adding the stock.
Make the sauce: Reduce heat to medium-low. Add heavy cream, Dijon mustard, and Asiago. Stir gently until the cheese melts and the sauce becomes smooth and slightly thickened. Avoid boiling.
Finish: Return the chicken and any resting juices to the skillet. Spoon sauce over the chicken and simmer gently for 3–5 minutes until heated through.
Garnish: Finish with fresh parsley and a light crack of black pepper.
