Preheat oven to 325°.
In an oven-safe skillet, melt butter over medium heat. Stir in olive oil and asparagus, and cook for 10 minutes or until the asparagus is tender.
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 to broil, and cook until cheeses are melted and lightly browned.