In a mixing bowl, combine the flour, sugar, salt, and cinnamon. Add cold, diced butter and use a pastry cutter or your fingers to mix until the mixture resembles coarse crumbs. Then, add water and vanilla extract, mixing just until the dough comes together. Form into a disc, wrap in plastic, and chill for at least 30 minutes.

Meanwhile, prepare the peaches. Slice them thinly and toss gently with lemon juice, a little sugar, cornstarch, and a pinch of cinnamon. Set aside to let the flavors meld and the juices release a bit, which will help thicken the filling during baking.

Preheat your oven to 375°F (190°C). Remove the chilled dough and roll it out on a lightly floured surface into a rustic circle about 12 inches wide. Transfer to a parchment-lined baking sheet.

Place the sliced peaches in the center, leaving a 2–3 inch border around the edges. Fold the edges of the dough over the peaches, creating a free-form crust that is pleated or overlapping as needed.

Beat the egg and brush it over the crust to give a beautiful golden-brown finish. Bake in the preheated oven for 40–45 minutes, or until the crust is flaky and deeply golden brown and the filling is bubbling. Let it cool slightly before serving to allow the juices to set and enhance the flavors.