Heat the olive oil and butter together in a large skillet over medium-high heat until melted and hot.
Add the sliced mushrooms and cook them slowly, stirring occasionally, until they turn a deep golden brown, about 4 to 5 minutes.
Stir in the minced garlic along with a pinch of salt and pepper, then cook for about 1 minute until the garlic becomes golden and fragrant.
Pour in the verjuice (or broth and lemon juice mixture), let it sizzle, and stir while scraping up any browned bits from the pan until most of the liquid evaporates, about 1 minute.
Add the broth, cream, and Parmesan, then stir well and lower the heat to medium so the sauce simmers gently without boiling.
Simmer for 2 to 3 minutes, stirring occasionally, until the sauce thickens slightly but is still smooth and pourable.
Stir in the thyme, taste, and adjust the seasoning with more salt and pepper if needed, then remove the skillet from the heat and serve right away over steak, chicken, pasta, or vegetables.