For the pie filling: Preheat the oven to 350 degrees F. Fit the pie crust into a 9-inch deep-dish pie plate, place a piece of parchment paper on top and fill with pie weights or dried beans.
Parbake the pie crust for 5 minutes, remove from the oven and cool.
Combine the peaches, cornstarch, brown sugar, granulated sugar, lemon juice, cinnamon, nutmeg and salt in a large bowl and toss to mix well. Transfer the peach mixture to the pie crust.
For the cobbler topping: In a food processor, combine the flour, butter, baking powder, baking soda, salt and 1 tablespoon of the granulated sugar, Pulse until the mixture is grainy with pea-size chunks of butter, about 30 seconds. Do not over process or the topping will become runny. Add the buttermilk and pulse 2 to 3 more times to combine.
Dot the topping over the top of the pie filling. Do not make the dots too big or they will not cook through. Sprinkle the remaining 1 tablespoon sugar over the top.
Bake the pie until the edges of the crust are golden brown, the cobbler topping is cooked through and the fruit is tender, 40 to 50 minutes. If the pie crust browns before the cobbler topping is cooked through, cover the edges with foil and continue baking.