Step 1: In a large bowl or food processor, combine the flour, 1 tablespoon of sugar, and salt. Add the cold, cubed butter and pulse or cut in until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 2: Gradually add ice water, one tablespoon at a time, mixing until the dough just starts to come together. Shape into a disk, wrap in plastic, and refrigerate for at least 1 hour.
Step 3: Preheat your oven to 400°F (200°C) and line a large baking sheet with parchment paper.
Step 4: In a medium bowl, toss the sliced peaches with 1/4 cup sugar, cornstarch, and vanilla extract until evenly coated.
Step 5: On a lightly floured surface, roll out the chilled dough into a 12-inch circle. Transfer the dough to the prepared baking sheet.
Step 6: Arrange the peach slices in the center of the dough, leaving a 2-inch border. Fold the edges of the dough over the peaches, overlapping the dough as you go to create a rustic border.
Step 7: Brush the folded edges of the dough with the beaten egg and sprinkle the entire galette (crust and fruit) with coarse sugar.
Step 8: Bake for 35-40 minutes, or until the crust is golden brown and the peach juices are bubbling. Allow to cool slightly before slicing.