Prepare the Fresh Peach Filling
Add the fresh peaches, granulated sugar, brown sugar, lemon juice, cinnamon, nutmeg and salt to a large saucepan.
Cook over medium heat for 6–8 minutes, stirring occasionally, until the peaches begin to soften and release their juices.
Mix the appropriate amount of cornstarch and water for your chosen crust.
Stir the slurry into the peaches. Cook for 1–2 minutes for the pie-crust version or about 30 seconds for the batter version.
Remove the saucepan from the heat. Stir in the vanilla and almond extract, if using.
Prepare the Canned Peach Filling
Drain the canned peaches, reserving 1 cup of heavy syrup.
Add the reserved syrup, brown sugar, cinnamon, nutmeg, salt and the appropriate amount of cornstarch to a large saucepan. Whisk until the cornstarch dissolves.
Cook over medium heat until the syrup begins to thicken. This should take 2–3 minutes for the pie-crust filling or approximately 1 minute for the batter version.
Fold in the drained peaches and cook for another 1–2 minutes.
Remove the saucepan from the heat. Stir in the vanilla and almond extract, if using.
Pillsbury Pie-Crust Method
Preheat the oven to 400°F. Lightly spray a 9-by-13-inch baking dish with nonstick cooking spray.
Unroll two pie crusts. Arrange them across the bottom of the dish, slightly overlapping the dough. Press the seam together and bring the crust slightly up the sides. Use excess dough to patch any uncovered areas.
Prick the bottom crust several times with a fork. Cover it with parchment paper and add pie weights or dried beans.
Bake for 10 minutes. Remove the parchment and weights, then bake for another 3–5 minutes. Remove the dish from the oven and reduce the temperature to 375°F.
Spoon the prepared fresh or canned peach filling over the crust. Cut the butter into small pieces and scatter it over the filling.
Cut the remaining two pie crusts into approximately 1-inch-wide strips.
Arrange half the strips in one direction over the filling. Weave the remaining strips over and under the first set to create a lattice top. Trim and seal the edges.
Whisk together the egg and water. Brush the egg wash over the lattice.
Combine the sugar and cinnamon, then sprinkle it over the crust.
Place the baking dish on a lined sheet pan. Bake at 375°F for 40–50 minutes, until the crust is golden brown and the filling is bubbling near the center.
Let the cobbler rest for at least 20 minutes before serving.
Classic Cobbler-Batter Method
Preheat the oven to 375°F.
Place the butter in a 9-by-13-inch baking dish. Put the dish in the oven for 5–7 minutes, just until the butter melts. Carefully remove it from the oven.
Whisk together the flour, sugar, baking powder and salt in a large bowl.
Add the milk and vanilla. Whisk just until the batter is smooth.
Pour the batter evenly over the melted butter. Do not stir.
Spoon the prepared fresh or canned peach filling evenly over the batter, including the thickened peach juices. Do not stir the layers together.
Combine the remaining sugar and cinnamon. Sprinkle the mixture over the top.
Place the baking dish on a lined sheet pan. Bake for 50–60 minutes, until the crust is deeply golden and completely set in the center.
Let the cobbler rest for at least 20 minutes before serving.