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Melt butter in a large non-stick pan over medium heat. Saute the apples for about 5 minutes until the are fully warmed throughout but not falling apart. Set aside to cool for a few minutes.
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Meanwhile, simmer the apple juice over medium-low heat. Continue to simmer until the volume of juice has reduced to 1 cup.
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Add the reduced juice to the apples and return to the stove top on medium heat to get the mixture boiling, then add the sugars, cinnamon and nutmeg.
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Simmer for only a minute before adding a thickening slurry made from dissolving together the corn starch and water.
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Stir continuously as you add the corn starch slurry and continue to simmer stirring constantly for about a minute. Remove from heat and allow to cool to room temperature before using to fill the bottom pie crust.
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Roll the first dough round into a 12 inch round and place in the bottom of a 10 inch pie plate. Push the dough into the corners and ensure that it is not stretched at all or it will shrink from the edge.
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Trim dough to edge of plate. Add the cooled apple pie filling.
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Roll the remaining dough round into 12 inch square-ish shape and cut that into 8 or 10 equal strips for the lattice crust top.
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Place the remaining pastry strips on top of the apple pie filling, interweaving them to form a simple lattice pattern.
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Press the ends of the strips into the edge of the bottom crust and trim the excess with a sharp knife.
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Brush the edges of the crust and the lattice top with an eggwash made by whisking together the egg yolk and water.
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Sprinkle the lattice top with turbinado sugar if desired.
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Chill the pie in the fridge for about 20 minutes to a half hour before placing it on a cookie sheet (to catch any drips from the filling) in a preheated 400 degree F oven. Bottom rack is best.
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Bake for 20 minutes at that temperature then reduce heat to 350 degrees F and bake for an additional 30- 35 minutes or until the crust is an even golden brown. The filling need not bubble for this pie to be done.
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Allow to cool on a wire rack at room temperature for at least a couple of hours before serving.
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NOTE: If your lattice crust edges begin to get too brown wrap them with aluminum foil for the last of the baking time.