In a large skillet, heat the olive oil and melt the butter over medium-high heat. Once the butter has melted, add the sliced mushrooms and allow them to cook leisurely until they are a rich golden brown, which should take about 4 to 5 minutes. Stir occasionally but avoid constant stirring to let the mushrooms develop a deep caramelized flavor.

As the mushrooms near golden perfection, sprinkle them with a pinch of salt and pepper, then add the minced garlic. Cook for another minute, stirring occasionally, until the garlic is fragrant and lightly golden.

Pour in the white wine, allowing it to sizzle as it hits the hot skillet. Stir the mixture, making sure to scrape up any browned bits from the bottom of the pan, and cook until most of the liquid has evaporated, approximately 1 minute.

Add the chicken or vegetable broth, heavy cream, and finely grated parmesan cheese. Stir to combine the ingredients, then reduce the heat to medium. Allow the sauce to simmer gently—avoid rapid boiling, as this could cause the cream to separate.

Continue to stir occasionally, letting the sauce simmer for 2 to 3 minutes until it thickens to a velvety consistency. The sauce should remain pourable and rich without being overly thick.

Stir in the fresh thyme leaves and adjust the seasoning with additional salt and pepper to taste. Remove the skillet from the heat.

This luscious mushroom sauce is perfect for draping over steak or other cooked proteins. It can also be tossed through pasta, spooned into baked potatoes, or served alongside roasted beef, chicken, pork, or even omelets. For more serving ideas, refer to the suggestions mentioned.