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 adhere any remaining pieces of dough to it, then wrap in plastic wrap or parchment paper. Refrigerate for at least 45 minutes before rolling out.
Center a rack in the oven and preheat to 350 degrees F. Take the galette dough out of the fridge and let it sit out while you prepare the strawberries.
In a medium bowl, combine the sliced strawberries, sugar, cornstarch, and vanilla bean paste. Gently stir until cornstarch disappears and the berries are well coated. Set aside.
Place the galette dough on a large sheet of parchment paper. Starting at the center of the dough, roll out, forming a 14-inch wide circle. Place the dough with parchment paper on a baking sheet.
Place the strawberry mixture on top of the dough, leaving a 2-inch border. Fold the dough border gently over the berries, overlapping where necessary and pressing gently to adhere the folds.
In a small bowl, beat the egg and water together with a fork. Lightly brush the edge of the dough with the egg wash and sprinkle turbinado sugar over the galette crust.
Bake the galette for 45-55 minutes, or until the crust is golden brown. Transfer the baking sheet to a wire rack to cool to room temperature before slicing.
Cut the galette into pieces and serve with vanilla ice cream or whipped cream, if desired.