Filling:

1. Slice 1 cup of cherries in half. Microwave for about 2 minutes in a covered glass dish. Mash well, pass mixture through a sieve. Save the strained cherry flesh and juice, discard the solids or save for oatmeal. Place 3 cups of whole cherries in a large bowl.

2. Mix white sugar, lemon juice, apple juice, brandy and salt until solids are dissolved. Pour liquid mixture over the cherries in a bowl. Let it sit about 2 hours, stirring occasionally.

3. Place apple flesh and lemon zest in a saucepan. Mix in strained mashed cherry flesh and juice. Drain the liquid from the bowl of whole cherries that have been soaking, and add liquid to the pan. Bring to a boil over medium heat. Lower heat; simmer until liquid is reduced in half. Remove from heat. Add almond extract. Let cool 3 minutes. Mix together tapioca flour and brown sugar in a medium bowl. Add cooked mixture to bowl and stir. Let cool to room temperature. 

Crust:

1. Mix together the flours then divide equally into 2 bowls. Add salt and sugar to bigger bowl and mix. Add butter to bigger bowl and cut in well using pastry blender. Add about 2/3 of the other flour to the dough, mix lightly. Mix together vodka, water, and vinegar. Add liquid to dough in 2 increments, mixing with hands until dough holds together. Divide equally in two; form into 6-inch discs, wrap each well in wax paper, and refrigerate at least 30 minutes.

Pie assembly:

1. Preheat oven to 425 degrees. Butter and flour a 9-inch glass pie plate. Roll out one pie dough disk onto a floured sheet of parchment paper to about a 12-inch circle. Fit into pie plate. Refrigerate.

2. Roll out one pie dough disk on floured parchment into a 10-inch by 14-inch oval. Place on flat cookie sheet in freezer. Beat together the egg white and 1 teaspoon of water. Brush onto bottom crust. Mix whole cherries together (3 cups soaked in juice/sugar/brandy and 1 cup unsoaked). Arrange into 1 1/2 layers on bottom crust. Pour cooked cooled mixture around whole cherries. Dot with 2 tablespoons of butter.

3. Remove oval crust from the freezer, slice dough oval into 1-inch wide strips the short way. Brush lightly with half-and-half. Weave a lattice atop the pie filling. Sprinkle with sugar and zest mixture. Crimp edges. Cut crust evenly with pie plate edge.

4. Place pie in center rack of 425 degree preheated oven. Bake for 15 minutes. Reduce heat to 350 degrees. Place foil shield on crust edge. Bake 30 minutes. Remove shield. Bake another 5 to 15 minutes until pie filling starts to bubble and crust is lightly browned. Remove from oven. Cool on a wooden cooling rack for 2 hours. Refrigerate 1 hour before serving.