Melt 2 tablespoons butter in a skillet over medium-high heat. Add chopped mushrooms, onion, and garlic. Season with salt, pepper, and thyme. Cook and stir until mushrooms are slightly softened, approximately 3 or 4 minutes. Transfer cooked mushrooms and onions to a plate and return pan to the heat. Add an additional tablespoon of butter to the same pan and when it has melted, add the orzo. Cook and stir for a couple of minutes until lightly browned. Stir in broth and bring to boil. Cover. Reduce heat and simmer until orzo is tender and liquid is absorbed, about 15 to 20 minutes.
Return cooked mushroom mixture to the skillet and stir to combine with the cooked orzo. Mix in ½ cup Parmesan cheese, the sour cream, and basil. Sprinkle remaining Parmesan cheese over the top. Taste and season with additional salt and pepper, if needed. Serve warm.