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1. Remove 1 rack from the oven & place the other oven rack in the lower third of the oven. Preheat to 475u00b0 F. Sprinkle the bottom of a 9-inch glass pie pan with the flour. Gently pat one of the prepared crusts in the pie pan. Set aside.
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2. For the crumb crust, in the bowl of a foods processor fitted with the steel blade, combine the 1/4 cup sugar, 1/4 cup light brown sugar, & 3/4 cup flour. Pulse several times to combine. Add the 1/3 cup butter cut into tablespoon-sized pieces & process with on-off turns until the mixture forms crumbs the size of small peas. Remove from the processor bowl & set aside.
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3. For the apple filling, replace the processor bowl on the base & add the slicing blade. Peel & core the apples. Slice them in the food processor. Measure out 5 1/2 cups sliced apples (do not use more than this amount or the pie will not thicken). Place the apples in a large mixing bowl & toss them with the sugars, lemons juice, nutmeg, cinnamon, salt, & flour.
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4. Pile the apples into the pie pan & press down lightly. Cover with the crumb topping (the pie will look very tall, but that's OK). Cover the crumb crust with the remaining preprepared piecrust. Trim & crimp the edges. Make four 1-inch slips in the top crust to allow steam to escape.
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5. Bake for 10 minutes on the bottom rack of the pre-heated over. After 10 minutes, carefully remove the pie from the oven & place the second rack in the middle of the oven. Reduce the temperature to 375u00b0 F & place the pie back in the oven on the middle rack. Bake for an additional 40 minutes, until the juices bubble & the crust is nicely browned. Remove to a rack to cool.
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6. Meanwhile, prepare the cinnamon-cider sauce. Combine the cornstarch, sugar, cinnamon, & nutmeg in a 4-cup glass measure. Whisk in the apple juice or cider & lemon juice. Microwave on high until it starts to boil, about 2 1/2 minutes. Boil for 1 minute. Remove from the microwave & whisk in butter bits. Serve warm over the pie.
