For the pie dough: Pour 1/4 cup water into a small container with 2 ice cubes.
Put the flour in a large mixing bowl. Add the granulated sugar and whisk to combine. Add the butter and shortening to the bowl and use a pastry cutter to work the fats into the flour mixture until pea-size lumps form. Sprinkle the ice water onto the mixture 1 tablespoon at a time, using your hands to pull the dough together, until the dough becomes just sticky enough to form a ball; you may not need all the ice water. Press the dough into a disc and wrap in plastic wrap. Refrigerate the dough to chill for a minimum of 1 hour. This step can be done several days in advance.
When ready to bake the pie, pull the dough out of the refrigerator and rest at room temperature for 15 minutes. Sprinkle the counter and rolling pin with additional flour. Preheat the oven to 350 degrees F.
For the filling: Set a 10-inch cast-iron skillet over medium-high heat. Add the butter and heat until melted. Cook until the bubbles subside and the butter is light brown and smells like toasted nuts, 5 to 7 minutes. Reduce the heat to medium and sprinkle the brown sugar over the butter. Add the cinnamon and salt and stir to combine. Cook until the sugar is dissolved, about 2 minutes more. Pour in the spiced rum and honey and stir until incorporated. Add the peaches to the sauce and sprinkle with the cornstarch. Carefully stir the peaches to coat in the brown butter sauce. Let cool for 5 minutes.
While the filling is cooling, roll the dough into a 12-inch circle that is about 1/8 inch thick. Gently lay the pie dough circle over the peaches, working carefully as the skillet will still be warm. If the dough is hanging over the edge, trim it with kitchen shears. Moving around the rim of the dough circle, tuck the edge of the dough into the skillet. Using a paring knife, vent the top of the pie by cutting a few small slits in the dough.
For the topping: Gently brush the dough with the beaten egg and sprinkle evenly with the demerara sugar. Bake until the crust is golden brown and the filling is hot and bubbly, 40 to 45 minutes. Let cool for 5 minutes. Serve warm with vanilla ice cream.