To make the dough, combine the flour, sugar, and salt in the bowl of a food processor. Pulse a few times to combine. Add in the cold butter cubes and pulse 8 to 10 times, until the largest pieces of butter are the size of small peas. Pour the ice-cold water into the flour mixture and pulse just until the dough comes together.
Transfer the dough to a piece of plastic wrap, form it into a ball, and flatten into a round disk. Wrap in the plastic and refrigerate while you prepare the filling.
To prepare the filling, place the sliced peaches in a large bowl and sprinkle on the cornstarch and 1 tablespoon sugar. Toss gently to combine.
Preheat oven to 350° F with a rack in the center of the oven.
Transfer the dough to a piece of lightly floured parchment paper. Roll it out to a 14-inch round. Transfer the parchment and dough to a rimmed baking sheet.
Stir the peaches again to make sure everything is well combined. Leaving any juices behind in the bowl, place the peaches into the center of the dough, leaving about a 3-inch border of dough all around. Carefully fold the edges of the dough up over the fruit to create a rim about 2 inches wide. Work your way around the dough, pleating the edges as you go.
Brush the dough with the milk and sprinkle with the 1 teaspoon sugar.
Bake the galette in the preheated oven for 50 to 60 minutes, until the dough is golden brown. Let the galette cool slightly before serving. Serve with vanilla ice cream or whipped cream, if desired.