Season both sides of the chicken breasts with salt and pepper.
Heat 1 tablespoon of olive oil in a large skillet over medium-high heat.
Cook the chicken for 5–7 minutes per side, until golden brown and fully cooked. Transfer the chicken to a plate and set aside.
In the same skillet, add the remaining olive oil and the butter.
Once the butter melts, add the sliced mushrooms. Sauté for 5–6 minutes until browned and tender.
Stir in the garlic and cook another 1–2 minutes until fragrant.
Pour in the chicken broth to deglaze the pan, scraping up any browned bits. Let it simmer for 2–3 minutes to reduce slightly.
Reduce the heat to low. Add the heavy cream, grated Asiago cheese, Dijon mustard, and thyme. Stir until the cheese has melted and the sauce is smooth and velvety. Taste and season with extra salt and pepper if needed.
Return the chicken to the skillet. Spoon the creamy sauce over the chicken and let it simmer for 3–5 minutes so the flavors meld and the chicken warms through.
Sprinkle with fresh parsley before serving. Pair with mashed potatoes, pasta, or roasted vegetables for a complete meal.