Preheat an oven to 425 degrees F (220 degrees C). Press one of the pie pastries into a 9-inch pie plate; reserve the remaining pastry.
Pour the lemon juice into a container that large enough for the apples and fill halfway with cold water. Peel and slice the apples into the lemon water; set aside.
Melt the butter in a large saucepan over medium heat. Stir in the flour and cornstarch to form a paste. Stir in 2 tablespoons of water along with the vanilla extract, 1/2 cup white sugar, brown sugar, 1 teaspoon cinnamon, and nutmeg; bring to a simmer. Drain the apples and add them to the sugar mixture along with the cranberries. Cook and stir 5 minutes; remove from the heat and allow to cool slightly.
Pour the fruit into the pie plate and press on the top crust. Stir together 1 tablespoon of white sugar and 1/2 teaspoon of cinnamon; sprinkle over the pie.
Bake in the preheated oven for 15 minutes; reduce the oven temperature to 350 degrees F (175 degrees C) and continue baking until the top crust is golden brown, 35 to 40 minutes.
