Preheat oven to 350. Melt butter in saucepan. Stir in flour to form a paste. Add water, white sugar, brown sugar, cinnamon, nutmeg, vanilla, and cornstarch, and bring to a boil. Reduce temperature and let simmer.

Place bottom crust in glass 9-in. pie plate. Brush egg whites on bottom crust to keep it from getting soggy. In bowl, mix 3/4 of sugar mix with apples. Mix well. Then pour coated apples into pie crust, mounded slightly. Cover with lattice work crust. Gently pour remaining sugar mix over crust.

Bake for about 45 to 50 minutes at 350. Brush pie with a little milk, and sprinkle with cinnamon and sugar. Continue baking 5 to 10 minutes longer, til apples are soft.

Let pie cool before serving. (so it's not too runny)