In a food processor, pulse flour, salt, and 2 tablespoons sugar until combined. Add butter.
Pulse until mixture resembles coarse meal, with a few pea-size pieces of butter remaining. Then, with machine running, add ? cup ice water.
Pulse until dough is crumbly but holds together when squeezed (if necessary, add up to ½ cup ice water, 1 tablespoon at a time). Do not overmix.
Divide dough into 2 disks and wrap each tightly in plastic. Refrigerate until firm, at least 1 hour (or up to overnight).
In a large saucepan or skillet - add your vanilla, lemon juice, sugars, cinnamon, and cornstarch. Wish well to combine. Continue to whisk for several minutes to allow your wet ingredients to thicken. This will help give your pie the filling the right consistency.
Slice the peaches into ¼-inch thick slices. You can use fresh or frozen, peeled or unpeeled peaches. Add the peaches to the saucepan and simmer for 5-7 minutes. This time will depend on how hard or soft your peaches are. Allow them to soften slightly, but not get mushy.
Allow the peach pie filling to cool to room temperature. If you are storing it for a future peach pie, then place the filling in glass jars and keep in the refrigerator for up to two weeks, or freezer for up to three months.
Lightly flour a work surface and rolling pin. Roll out 1 disk dough to a 12-by-16-inch rectangle. Roll dough over rolling pin; carefully unroll into a 10-by-14-by-1-inch rimmed jelly-roll pan. Press gently to fit into pan.
Pour in peach filling, then lightly brush edges of dough with water. On floured surface, roll out remaining disk to an 11-by-15-inch rectangle. Roll dough over rolling pin and carefully unroll over filling.
Press along moistened edges to seal. Fold under any overhang, tucking it into the pan. With your fingers, crimp edges.
With a paring knife, cut slits on top to vent.