For the Pie Dough: In the bowl of a kitchen mixer, add the flour, sugar, salt, and butter. Pulse on/off to blend. Add the water gradually, a little at a time, until the dough just comes together but is not overly wet.

Divide the dough in half. Using your hands, flatten each half into a 1-inch thick disc.

Wrap each disc tightly in plastic wrap and refrigerate for at least 2 hours.

For the Pie Filling: Adjust the oven rack to the lower third position Preheat oven to 425 degrees F. Line a baking sheet with aluminum foil and set aside.

In a large bowl, add the sugar and cornstarch, and mix to blend. Next, gently fold in the raspberries, blackberries, blueberries, and vanilla extract until evenly coated. Set aside while you prep the pie crust.

On a floured work surface, roll out one of the discs of chilled dough. Roll half the pie dough into a 12-inch circle. Place the dough into a 9-inch round pie dish. Tuck the dough in slightly under itself with your fingers, making sure it's smooth. Pour the filling into the pie dish. Set aside while you work on the top pie crust.

For the Lattice: Remove the other disc of chilled pie dough from the refrigerator. Roll the dough into a circle that is 12 inches in diameter. Using a sharp knife, cut the dough into ten (1-inch-wide) strips. Carefully weave the strips over and under one another, pulling back strips as necessary to weave. Press the edges of the strips into the bottom pie crust edges to seal. Use a small paring knife to trim off any excess dough. Flute or crimp the pie crust edges with a fork as desired.

Using a pastry brush lightly brush the top of the pie crust with the egg wash. Sprinkle the top with coarse sugar, if using.

Place the pie on the prepared baking sheet, place in the oven, and bake at 425 F for 25 minutes, then, reduce the oven temperature down to 375 degrees F. Place a pie crust shield (or foil) on the edges to prevent it from over-browning. Continue baking the pie until the filling’s juices are bubbling, about 40–50 more minutes.