. To make the dough, combine flour, sugar, cornmeal, and salt in a processor; pulse 2-3 times. Add butter and pulse 4-5 times, or until mixture resembles coarse meal. With the processor running, slowing pour the buttermilk through the chute, processing until the dough forms a ball. Remove the dough ball and pat it into a firm ball. Wrap in plastic wrap and refrigerate for at least 45 minutes before rolling out.
2. Center a rack in the oven and preheat to 350 degrees F.
3. Remove the dough from the refrigerator and place on a sheet of parchment paper. Cut the dough into four equal parts. Starting at the center of the dough, roll out each piece of dough into a circle, about 6 inches. Place the dough with parchment paper on a baking sheet. Arrange mixed berries in the center of each galette, leaving a 1-inch border. Fold the border over the berries, overlapping where necessary and pressing gently to adhere the folds.
4. Lightly brush the edge of the dough with the egg and sprinkle crust and berries with turbinado sugar.
5. Bake the galettes for 40-45 minutes, or until the crust is golden brown. Transfer the baking sheet to a wire rack to cool for at least 20 minutes. Serve warm or at room temperature.
*Note-you can make the galette dough in advance. It will keep in the refrigerator for 3 days. Serve with whipped cream or ice cream if you wish.