Bring a large pot of water to a boil over high heat. Fill a large bowl with ice water. Add the peaches to the pot and cook for about 1 minute. Using a slotted spoon, transfer the peaches to the bowl of ice water; let cool for 2-3 minutes.

Transfer the peaches, open ends down, to a paper-towel-lined plate. Peel the skin off of the peaches and slice them into ¾-inch wedges. Transfer the slices to a second large bowl. Add the granulated sugar and toss to combine. Let sit at room temperature for 30 minutes.

Strain the peaches over a medium skillet, letting all the peach juice drain into the pan, then return the peaches to the bowl. Heat the skillet over medium heat. Add in the vanilla, lemon juice, nutmeg, cinnamon, and salt and stir to combine. Sprinkle the cornstarch over the peach juice mixture and whisk to combine. Continue whisking until the sauce has thickened to a paste-like consistency, 2-3 minutes. Remove the pan from heat, then add the peach slices and, using a rubber spatula, gently fold to combine. Let cool in the pan.

Preheat the oven to 400°F with a rack in the center position.

Lightly dust a clean work surface with flour. Working with one piece of dough at a time, roll the dough into two 14-inch rounds (about ¼ inch thick). Gently press one dough round to a 9-inch pie plate without stretching it. Trim the edges if necessary. Add the cooled peach mixture on top of the dough and gently spread into an even layer. Lay the second dough round over the filling and crimp the edges of the two dough rounds together with a fork or your fingers to seal.

Brush the top of the pie with the beaten egg and sprinkle with the Demerara sugar, if using. Using a sharp knife make two, 3-inch slits in the top of the crust in an X shape.

Place the pie plate on a baking sheet and bake on the center rack for 15 minutes. Reduce the oven temperature to 350°F and continue to bake until the crust is a deep golden brown and the filling is bubbling, about another 70 minutes. If the crust is browning too quickly, tent the pie with foil halfway through baking.

Let the pie cool completely, about 2 hours, before cutting. Serve with ice cream or whipped cream.