Heat the butter and olive oil in a large skillet over medium-high heat.
Add the sliced mushrooms to the skillet. Cook without stirring for 3 to 4 minutes until they begin to brown. Stir and continue cooking until the mushrooms release their liquid and turn golden brown, about 5 to 7 minutes total.
Add the minced garlic and dried thyme to the skillet. Cook for 1 minute until fragrant.
If using wine, pour it into the skillet and scrape up any browned bits from the bottom. Let the wine reduce by half, about 2 minutes.
Pour in the broth. Bring the mixture to a simmer and cook until the liquid has reduced by about one-third, concentrating the flavor, approximately 5 minutes.
Stir in the heavy cream and Dijon mustard. Reduce the heat to medium-low and let the sauce simmer gently until it thickens enough to coat the back of a spoon, about 3 to 5 minutes. Do not boil.
Season the mushroom sauce with salt and pepper to your preference.
Remove from heat. Stir in half of the fresh parsley. Serve immediately over steak, pasta, or chicken, garnished with the remaining parsley.
Notes
For a richer flavor, substitute half of the heavy cream with cream cheese, softened.
If you skip the wine, add a splash of extra broth or water when deglazing the pan.
This sauce tastes best when made fresh, but you can store leftovers in an airtight container in the refrigerator for up to 3 days. Reheat gently on the stovetop.