Whisk a small egg until well blended. Set aside. You will use only 1 tablespoon of the egg and the rest you can discard or use for breakfast.
For The Cheddar Cheese Butter Pastry
Cut the cold butter into small cubes. Divide the cubes into two parts, about two thirds to one third. Wrap each portion with plastic wrap. Refrigerate the large portion and place the smaller portion into the freezer for at least 30 minutes.
Place the flour, baking powder, and salt in a resealable bag, shake the bag to combine and place the bag into the freezer for at least 30 minutes.
Food Processor Method
Place the contents of the flour bag into a food processor fitted with the metal blade. Add the butter cubes from the refrigerator and pulse until it resembles coarse meal. Add the frozen butter cubes and pulse until the frozen butter is the size of small peas.
Add the vinegar and egg and pulse 3 - 5 times.
Add the water, one tablespoon at a time, pulsing between each addition. Test the dough after tablespoon #5. The dough will begin to hold together when pinched between your fingers. If it doesn't hold together when pinched, add another tablespoon of water and pulse again. Note that the dough will look like coarse sand and will not stick together without pinching so you may think that you need more water, but you don't. Test the dough before each addition of water. Too much water will result in a tough dough.
Transfer the dough to a clean work surface. Add the cheddar cheese and working quickly, shape the dough into two equal sized discs, wrap each with plastic wrap and refrigerate for 30 minutes.
Pour the flour mixture into a large bowl. Cut in the refrigerated butter with a pastry blender or two knives until the mixture resembles coarse meal. Add the frozen butter cubes and cut in until the frozen butter is the size of small peas.
Sprinkle the vinegar and egg over the flour. Gradually add the ice water one tablespoon at a time, mixing quickly after each addition until it sticks together. Add the cheddar cheese and shape the dough into two equal sizes discs, wrap each with plastic wrap and refrigerate for 30 minutes.
For The Apple Pie
Preheat the oven to 400 F.
Fill a large bowl with water and lemon juice and set aside.
Peel and core each apple and slice into 1/4" thick slices. To prevent browning, put the slices into the water/lemon bath until all the apples are done and the pie is ready to be assembled.
Remove one disc of dough from the refrigerator and on a lightly floured surface roll the dough into a circle that is about 1" larger than your pie plate. Transfer the dough to the pie plate.
In a bowl, whisk together the sugar, cinnamon, flour, and cornstarch. Add the apples and gently stir to coat.
Begin by assembling the apples in layers and continue until all the apples are used. If there is any sugar mixture and juices left in the bowl, simply pour it over the top of the apples. Dot the surface of the fruit with butter cubes.
Remove the remaining disc of dough from the refrigerator and on a lightly floured surface roll the dough into a circle that is large enough to fit over the mound of fruit with an additional inch all the way around. Use a pastry brush to add some of the whipping cream along the edge of the bottom crust layer to act as a "glue". Transfer the dough for the top crust to the pie.
Use your hands to press the edges together lightly. Roll up any of the overhanging dough to form the crust along the edge of the pie. Use your fingers to pinch the dough into a simple crimped edge crust pattern.
Cut a small hole in the top center of the upper crust for ventilation. Use a pastry brush to brush the top and edges of the pie with whipping cream. Sprinkle with additional cheese and coarse sugar if desired.
Bake for 1 hour to 1 hour 15 minutes or until golden on top. Remove from the oven and cool on a wire rack.