Heat butter and olive oil in a large skillet over medium heat.
Add onion (or shallots) and sauté 3–4 minutes until softened. Stir in garlic and cook for 1 minute until fragrant.
Add mushrooms, salt, pepper, and thyme. Cook 6–8 minutes, stirring occasionally, until mushrooms are golden and their liquid has mostly evaporated.
Pour in broth, scraping up any browned bits from the bottom of the pan. Simmer for 2 minutes.
Stir in heavy cream and bring to a gentle simmer. Cook 5–6 minutes until sauce thickens slightly.
Stir in Parmesan cheese until melted and smooth.
Remove from heat and sprinkle with parsley before serving.