Heat four teaspoons olive oil in a ten inch skillet with and oven proof handle over medium heat. Add the potatoes, and cook until brown, about six minutes, flipping after three minutes. Add onion, mushrooms and tomatoes and cook ten more minutes until everything is very soft. Remove pan from heat and allow to cool while you heat the broiler. Whisk the eggs, egg whites, water, salt and pepper together.
Add remaining two teaspoons oil to the pan and heat over medium. Add garlic and half of the basil and cook 30 seconds. Smooth vegetables into an even layer and pour eggs on top. Reduce heat to medium low and cover mostly, but not completely. Cook six to eight more minutes. Remove lid and sprinkle cheese evenly over the top. Place pan in broiler for 45 to 60 seconds to finish cooking the eggs. Remove pan and let stand for one minute. Slice frittata into four pie shaped slices. Use a spatula to remove the slices and place on plates. Garnish with remaining basil.