Using a large saucepan, add the apples, water, regular and brown sugars, lemon juice, cinnamon and nutmeg. Place over high heat and bring to a boil. Once boiling, reduce to a simmer and cook, uncovered, for 35 minutes, stirring occasionally.
Remove from the heat and stir in the vanilla. Allow to cool slightly.
Once cool enough to handle, place in a kitchen mixer, such as a Cusinart, fitted with a blade attachment and puree until smooth.
Pour into a large glass jar and store in the refrigerator. This applesauce will keep for 4 days.