In a large bowl, whisk together the all-purpose flour and salt. Using a pastry cutter or your fingertips, work the chilled, cubed butter into the flour until the mixture resembles coarse crumbs with pea-sized butter pieces.
Gradually sprinkle the ice water over the flour mixture, mixing with a fork just until the dough begins to clump together. Divide the dough into two balls, flatten each into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes.
Preheat your oven to 375°F (190°C). In another large bowl, whisk together the granulated sugar and cornstarch. Gently toss the mixed berries with the lemon juice and vanilla extract, then fold them into the sugar mixture until evenly coated.
On a lightly floured surface, roll out one disk of dough into a 12-inch circle and carefully transfer it to a 9-inch pie plate. Spoon the berry filling into the crust, then divide it into two sections. Add 2 drops of blue food coloring to one section and 2 drops of pink food coloring to the other, gently swirling each with a spoon to create a marbled effect.
Roll out the second disk of dough and cut it into 1-inch wide strips. Create a lattice crust over the filling by weaving the strips over and under each other. Trim any excess dough, crimp the edges, and brush the lattice with the beaten egg wash. Sprinkle generously with sparkling sugar.
Bake the pie for 45-50 minutes, or until the crust is deep golden brown and the berry filling is bubbly. Place the pie on a wire rack and allow it to cool completely, for at least 3 hours, before slicing to allow the filling to set.