1. 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.
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. Starting at the center of the dough, roll out, forming a 14-15-inch wide circle. Place the dough with parchment paper on a baking sheet. Generously sprinkle apple slices with cinnamon and sugar. Arrange apple slices in a spoke pattern on the galette dough, leaving a 2-inch border. Sprinkle the toffee bits over the apple slices, making sure all of the slices are covered. Fold the border over the apples, overlapping where necessary and pressing gently to adhere the folds.
4. Lightly brush the edge of the dough with the egg and sprinkle crust with turbinado sugar.
5. Bake the galette for 45-50 minutes, or until the apples are tender and 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.