1. Toast the almonds: Preheat the oven to 300°. Scatter the almonds in a single layer on a baking sheet and bake, stirring occasionally, until golden brown, about 15 minutes. Remove from the oven and cool completely.
2. Make the pastry: Process the almonds in a blender or food processor until finely chopped. Add the flour, granulated sugar, and salt and pulse to combine. Add the butter and pulse until it resembles small peas. Add the ice water, 1 tablespoon at a time, and almond extract, and process just until the dough comes together and forms a ball around the blades. (Do not overprocess.) Remove the dough from the processor, wrap it in plastic wrap, and chill at least 30 minutes or overnight (or freeze up to 2 months).
3. Place a rack in the upper third of the oven and preheat to 375°.
4. Make the filling: Place the peach slices in a large bowl. In a small bowl, stir together the granulated sugar, cornstarch, and cinnamon until well blended. Add the sugar mixture to the peaches, stirring gently.
5. Turn the dough out onto a floured pastry cloth or surface and roll into a 16-inch circle using a floured rolling pin. Fold it in half, transfer to an ungreased 10-inch cast iron skillet, and gently unfold the pastry, fitting it into the bottom of the pan and allowing the excess pastry to hang over the edge. (Don't bother to trim the dough—any unevenness adds to the rustic quality of the tart.)
6. Spoon the peach mixture into the pastry, mounding it in the middle. Gently fold the edges of the pastry up around the filling, overlapping them in soft folds. Take care that the pastry doesn't tear around the edge of the tart or the juices will escape during baking.
7. Whisk together the egg and water in a small bowl. Brush the egg wash over the pastry and sprinkle it with the turbinado sugar. Bake until the pastry is golden brown and the fruit is hot and bubbly, 35 to 40 minutes. Let cool for 1 hour to set the juices and serve warm