Food processors make pie crusts a breeze, but they're easy to mix by hand as well.
Combine the flour, sugar, mustard, salt, and cayenne in a mixing bowl. Whisk lightly to mix.
Cut in the butter until a pea-sized meal is formed, then add the cheese and evenly mix.
Working with about 1 tbs at a time, work in the water. You're looking for dough that will hold together without crumbs when you squeeze it. You may need between 2 tbs and 1/2 cup, depending on a few factors, so don't fret if it's taking more than expected.
Divide the dough into two halves, then roll into a ball. Wrap in plastic wrap, then shape into a disk. Refrigerate at least 30 minutes, but up to 2 days.
Preheat your oven to 425 F and grab a 9-inch pie pan and a baking sheet large enough to hold it. The baking sheet will catch any spillage from the pie and help keep the oven clean.
While the dough is chilling, peel and core the apples. Slice them into 1/4 inch wedges, then transfer them to a mixing bowl.
Add the lemon juice and toss. This will help prevent them from browning.
Add the sugar, brown sugar, flour, allspice, and salt, then toss again.
Once the dough is ready, take one disk out and roll it out to about 12 inches.
Transfer it to the pie pan, then spoon in the apples. Dot the top of the apples with the small chunks of butter.
Roll out the remaining dough disk to about 11 or 12 inches, then lay it over the pie. Trim the crust to get 1 inch of overhang, then fold the top crust over the bottom crust. Press to form a seal, then crimp if desired.
Lightly beat the egg with 1 tbs water, then brush over the pie. Using a sharp paring knife, make 3 or 5 vent holes to help the steam escape.
Place the pie on the baking sheet, then bake for 25 minutes. This will help keep the crust crunchy instead of soggy.
Lower the temperature to 350 F and bake for 45 minutes.
Remove from oven and cool to room temperature before slicing. This will be hard because it takes about 3 hours, but it's worth it.