Season both sides of the chicken breasts with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat.
Add the chicken breasts to the skillet and cook for 5-6 minutes on each side, or until golden brown and cooked through. Remove chicken from skillet and set aside.
In the same skillet, add the sliced mushrooms and cook for 5 minutes, stirring occasionally, until they release their moisture and start to brown.
Add the chopped onion and minced garlic to the mushrooms and cook for 2-3 minutes until the onion is translucent and fragrant.
Pour in the chicken broth and stir, scraping the bottom of the skillet to loosen any browned bits. Let it simmer for 3 minutes to reduce slightly.
Reduce heat to medium-low and stir in the heavy cream, Dijon mustard, and dried thyme. Cook for 3-4 minutes until the sauce thickens slightly.
Return the cooked chicken breasts to the skillet, spooning some sauce over the top. Simmer for 2-3 minutes to warm the chicken through and meld the flavors.
Sprinkle chopped fresh parsley over the chicken and sauce before serving.