For the Crust:

1. Butter Prep: Grate the butter using a grater if NOT using a food processor. If you're using a food processor, cut into large cubes first. Or, if you're a traditionalist, you can wait to cut the butter into the flour using a pastry cutter or two butter knives.
2. In a stand up mixer, add the flour, sugar and salt. Add grated butter and mix until roughly incorporated in the flour. If using a food processor, pulse the flour, sugar, and salt, until well mixed. Pulse until pieces are pea size.
3. Slowly add the ice water, a tablespoon or two at a time, mixing/pulsing after each addition, until the dough just begins to form.
4. Turn out 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. Preheat the oven to 425°F.
7. After the hour has passed, place the plum slices in a bowl and sprinkle with the flour and sugar, then add the extract. Toss gently to get everything coated.
8. In a small bowl, whisk the egg and set aside.
9. 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.
10. Arrange the fruit slices in an overlapping pattern in a single layer in the center of the dough, forming about a 7 or 8-inch circle.
11. Taking the 1 tbsp of butter, break apart and scatter across the plums.
12. Fold the outer edges of the dough round over the filling.
13. Use a pastry brush to coat the exposed dough with an egg wash and sprinkle with coarse sugar if using.
14. 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.