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 the salt, pepper, and heavy cream. Cook, stirring occasionally, until the cream has thickened, about 5 minutes. If the sauce thickens too quickly, add water, one tablespoonful at a time, to thin. Turn off the heat and stir in the butter, egg yolks, and half 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.