Melt butter in an oven-safe skillet over medium heat. Stir in olive oil and asparagus, and cook until the asparagus is tender, about 10 minutes. Stir in the mushrooms, and continue cooking about 5 minutes.
In a medium bowl, whisk together eggs, water, and thyme. Pour into the skillet, and reduce heat to low. Cover, and cook 5 minutes.
Transfer the skillet to the preheated oven. Bake 10 to 15 minutes, until eggs are no longer runny. Top the mixture with Parmesan cheese and mozzarella cheese. Turn on the broiler, and broil until cheeses are melted and lightly browned.