Place flour, 2 tablespoons maple syrup, and salt into the bowl of a food processor. Pulse several times to combine. Add butter and pulse a few more times until mixture has the texture of coarse sand.

Mix cold water and vinegar together in a small measuring cup. Add 1/2 the mixture to the food processor; pulse a few times. Add additional mixture until dough comes together slightly. Divide dough in 1/2. Wrap each 1/2 in plastic wrap and form into discs. Refrigerate at least 1 hour.

Preheat the oven to 375 degrees F (190 degrees C). Line a large baking sheet with aluminum foil and place a 9-inch pie pan on top. Set aside.

Mix sour cherries, blackberries, blueberries, and ground cherries together in a bowl. Whisk 1 cup maple syrup, cornstarch, lemon juice, vanilla extract, cinnamon, almond extract, nutmeg, and salt together in a small bowl until smooth. Pour over berries and toss to combine; set pie filling aside.

Remove 1 disc of pie dough from the refrigerator. Roll out into a 12-inch circle on a lightly floured surface. Place evenly into the bottom and over the sides of the pie pan. Roll out second piece of dough into another 12-inch circle; cut into 1-inch strips. Pour pie filling into the crust and use dough strips to form a lattice pattern on top. Remove any excess overhanging dough and crimp the edges to seal.

Whisk egg and water together in a small bowl until combined. Lightly brush egg wash over pie crust. Cover edges of pie crust with aluminum foil to prevent overbrowning.

Bake in the preheated oven for 35 minutes. Remove foil and continue baking until crust is golden and filling is bubbling, 30 to 40 minutes more. Remove from the oven and let cool completely before serving.