To make the galette dough: pulse the flour, sugar and salt together in a food processor fitted with the blade attachment until combined. Add the butter and pulse until the mixture resembles coarse crumbs. Drizzle egg and pulse until a ball just begins to come together.
Form dough into a disk. Wrap in plastic and chill until firm, at least 2 hours.
Once chilled, remove the dough on a lightly floured surface. Cover a baking sheet/tray/dish with parchment paper. Either roll the entire dough out and transfer onto the parchment paper or roll out sections at a time and build the galette crust directly on the parchment paper, using your hands to pinch the pieces together. Chill back in the fridge while you prepare the berry filling.
Preheat oven to 400 F degrees. Toss stone fruit slices with sugar, cornstarch, lemon juice and zest, vanilla, and salt. Arrange the slices on the dough and fold the sides of the dough over.
Melt 1 tablespoon of the butter and brush the sides of the dough with it. Cube the other tablespoon of butter and scatter over the fruit filling. Sprinkle all with sugar.
Bake galette until the fruit is soft and cooked through, the fruit juices are bubbling, and the crust is a deep golden-brown, 40-50min.