Whisk together egg whites, flour and cold water until you get a smooth batter. I would add some salt, pepper, and garlic salt into batter. Add a bit more water to thin out if you don't want a thick batter coating.
Add vegetable oil into a pot or pan over medium high heat until it is hot enough to fry a drop of your tempura batter dropped inside.
Dredge 1 mushroom at a time in batter and put into hot oil. Flip and make golden brown on all sides. Then remove and put on to a few paper towels so it can absorb the excess oil on the outside.
Sprinkle with shredded parmesan cheese, rosemary, salt and pepper before serving.