1. Preheat the oven to 425°F.

For the Crust:

1. Grate the cold butter on a cheese grater and set aside.  
2. To the bowl of a stand up mixer, add the flour, sugar and salt. Add grated butter and mix until roughly incorporated in the flour.
3. Slowly add the ice water; a tablespoon or two at a time, mixing after each addition, until the dough just begins to form. Stop mixing once the dough is formed. 
4. Turn out the dough onto a clean surface. Use your hands to form into a disc. Work the dough only enough to bring it barely together into a disk.
5. Sprinkle with a little flour on all sides, wrap with plastic, and place into the refrigerator to chill for an hour.
6. Place the peach slices in a bowl and sprinkle with the flour and sugar, then add vanilla. Toss gently to get everything coated.
7. In a small bowl, whisk the egg and set aside.
8. On a lightly floured surface, roll out the dough to about a 12-inch diameter. Gently lift up the rolled out dough and place it on a rimmed baking sheet lined with parchment. 
9. Arrange the peach slices in an overlapping pattern in a single layer in the center of the dough, forming about a 7 or 8-inch circle. 
10. Break apart the 1 tbsp of butter and scatter across the peaches. 
11. Fold the outer edges of the dough round over the filling.
12. Use a pastry brush to coat the exposed dough with an egg wash and sprinkle with coarse sugar if desired.
13. Bake for about 15-20 minutes, until lightly browned. Remove from the oven and let cool on the baking sheet, over a rack, about 15 minutes. Serve with vanilla ice cream or fresh whipped cream.