In a large bowl, combine the flour, granulated sugar, and salt. Add the cold, cubed butter and cut it into the flour mixture using a pastry cutter or your fingers until the mixture resembles coarse crumbs.
Gradually add the ice water, 1 tablespoon at a time, and mix until the dough comes together. Form the dough into a disk, wrap it in plastic wrap, and refrigerate for at least 30 minutes.
In a large bowl, combine the mixed berries, granulated sugar, cornstarch, lemon juice, lemon zest, and vanilla extract. Gently toss to coat the berries evenly.
Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper.
On a lightly floured surface, roll out the chilled dough into a 12-inch circle. Transfer the dough to the prepared baking sheet.
Spoon the berry mixture onto the center of the dough, leaving a 2-inch border around the edges.
Fold the edges of the dough over the berries, pleating as necessary, to create a rustic edge. Brush the crust with the beaten egg wash and sprinkle with coarse sugar if desired.
Bake in the preheated oven for 40-45 minutes, or until the crust is golden brown and the filling is bubbly.
Allow the galette to cool on the baking sheet for 10-15 minutes before slicing and serving.