A proof

I like math.
I also like pie.
Therefore, I love National Pi Day.

QED.

Pi-Inspired Apple Hand Pies
A splendid mash up of recipes from Cooks' Illustrated (the glorious apple filling), Smitten Kitchen (just for having the idea of making hand pies.  Brilliant!), The Big Sur Bakery Cookbook and my dessert class (the delectable crust).  And of course, the number Pi, in whose honor I used a perfectly circular cookie cutter to make these.  There are quite a few steps involved here, but the end result is totally worth it. Kind of like a good math proof*.

For the crust:

3 c. flour
1/2 c. sugar
3/4 tsp. salt
zest of one orange
12 T. (1 1/2 sticks) butter, frozen and then grated finely
1/3 c. cold apple cider, plus extra as needed
1 tsp. vanilla

In the bowl of a food processor, combine flour, sugar, salt and orange zest by pulsing a few times.   Sprinkle frozen grated butter on top, then pulse again until mixture resembles coarse, sandy crumbs.   Combine apple cider and vanilla, then with the food processor running, slowly pour in the liquids.  Pulse just until the dough begins to come together into one or more big chunks.  There will still be some stray dry parts, and it should definitely not be one smooth ball.  If needed, add a little more apple cider, 1 tsp. at a time.

Turn crust out onto a smooth, cool surface, and compress with your fingers, incorporating any dry crumbs as you go.  Form into two discs, about 4-5 inches across and an inch high, place into a plastic bag, and refrigerate for an hour (or as long as a few days).

Working with one disc at a time (the other one stays in the fridge), roll dough out between two sheets of wax paper, until it's about 1/4 inch thick.  Using a 4 inch diameter** cookie cutter, make as many rounds as you can out of the dough.    I was able to make 24 rounds, but if you aren't nibbling on the scraps as you go like I was, it's possible you might get a few more or less.  No matter what, you need an even number of rounds.  Lay the rounds out between sheets of wax paper on a cookie sheet, and refrigerate for another 30 minutes. While those are chilling, make the apples.

For the apples:

1 c. apple cider or fresh apple juice
2 Tbs. fresh lemon juice
1/3 c. brown sugar
3 large or 4 medium Pink Lady apples, peeled, cored, quartered, and very thinly sliced
1/3 c. sugar
1 tsp. cinnamon
2 Tbs. butter
1 egg, beaten
Sugar crystals

In a small saucepan, cook apple juice over medium high heat until reduced to 1/2 c.  This will take about 10 minutes, more or less.  Add lemon juice and brown sugar, and cook over low heat, stirring every once in a while, for another 5 minutes.  Cool.

I used a mandoline to slice my apples, and it worked like a charm!  But a regular sharp knife is perfectly fine.  Remember to keep a little extra lemon juice handy as you are prepping the apples.  If you sprinkle some onto the cut slices as you go, they will not turn brown. Combine sugar and cinnamon, then pour over the sliced apples in a bowl.   Set aside.

In a large skillet, heat butter over medium high heat until sizzling.  Add the apple/sugar/cinnamon mixture, and cook, stirring often, for about 6-8 minutes, or until apples are warm but not cooked completely.  Remove from heat and stir in the apple juice mixture.  Eat a couple, just to see how delicious this is going to be in your pies.  Now stop.  Really.

Get the crust rounds out of the fridge.  Using an itty bitty cookie cutter, cut a hole in the center of 1/2 of the rounds.  These will be the tops of your pies.    Using a tablespoon or your hands, place a small mound of the apple mixture onto the bottom rounds, leaving 1/2 inch of crust around the apples so you can seal the pies.    Don't worry if you have some apples left over (you can eat them), and if there's a good amount of sauce left in the pan (we'll come back for that.)

Using your fingers or a pastry brush, spread a little of the egg wash around the filling on each bottom round.  Hang onto that egg wash...you'll need it again later.  Carefully place a top round over the filling, then work around the edges, pressing gently with the tines of a fork to seal the top and the bottom rounds together.   Using a small spoon, drizzle a little of that leftover pan sauce into the top hole of each pie.  Some will definitely run down the sides, and make a little mess.  Leave it!  These drippings will add nice crispy caramel bits to the pies when they bake.  

Pop the tray into the fridge for another 20 minutes while you preheat the oven to 375.

Take the pies out of the fridge, brush the tops lightly with some of the remaining egg wash, and sprinkle with sugar crystals.  Bake for 20-25 minutes, or until crusts are deep golden brown.   Cool for at least 15 minutes, then serve warm or at room temperature.   You truly can hold one of these in your hand and eat it just like that.

Yum. Yum. Yum.***

* Here's an example of a bad proof, which I actually had to do in a math class in college:  Prove that 0 is less than 1.  Show your work.  The proof took two pages, single spaced, and I changed my major the next day.

** That's 2(pi)r, for those of you following along at home.

*** I'm quoting here.