Add the mushrooms, toss to coat, and cook, stirring occasionally, until the mushrooms are very tender and a bit golden, 4 to 5 minutes. Reduce the heat to medium-low and add 1/4 teaspoon salt, 1/4 teaspoon pepper, and the heavy cream. Cook, stirring occasionally, until the cream has thickened, about 5 minutes. If the sauce thickens too quickly, add 2 to 3 tablespoons of water. Turn off the heat and stir in the butter, egg yolks, and 1 tablespoon of the Parmesan cheese. Add the zucchini noodles, then toss for 2 to 3 minutes to coat and warm the zucchini. Season to taste with salt and pepper and serve topped with the remaining Parmesan cheese.