Heat olive oil in a large pot over medium heat. Cook half the mushrooms without stirring to allow one side to brown. Flip them over, season with salt and pepper, and cook until golden brown. Set aside and repeat with the remaining mushrooms.

In the same pot, melt 1 tablespoon of butter. Add the finely diced onion and sauté until soft and lightly browned, about 5 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.

Pour in the dry white wine and let it simmer for about 5 minutes. Scrape the bottom of the pot to collect any flavorful bits and allow most of the wine to evaporate.

Add orzo, cooked mushrooms, room-temperature heavy cream, and chicken broth to the pot. Stir in the fresh thyme, parsley, salt, and black pepper. Bring to a gentle simmer over low heat, stirring frequently until the orzo is tender and the sauce thickens, approximately 15 minutes.

Remove from heat and mix in the remaining tablespoon of butter and freshly grated Parmesan cheese. If the sauce is too thick, add a bit more chicken broth to reach your desired consistency. Serve warm, topped with extra Parmesan and parsley if desired.