Preheat your oven to 375°F (190°C). Lightly grease a large 9x13-inch baking dish or a deep cast iron skillet to prepare it for layering.
In a spacious bowl, combine the sliced peaches with granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, lemon juice, and vanilla extract. Gently stir until the fruit is evenly coated and a syrup begins to form.
Roll out one of the pie crusts to fit the bottom of your prepared baking dish. Lay it flat and press gently into the base. Use a pastry brush to apply a thin coating of melted butter.
Spoon half of the prepared peach mixture over the crust. Distribute several pieces of butter over the surface of the fruit.
Unroll the second pie crust and lay it over the fruit. Don't worry about a perfect fit—uneven edges are fine. Brush with more melted butter and lightly scatter granulated sugar over the top.
Add the remaining peach mixture on top of the second crust. Once again, place small cubes of butter evenly across the fruit layer.
Position the third pie crust on top of the cobbler. Cut small vents or decorative shapes into the dough to allow steam to escape. Finish with a brush of melted butter and a dusting of granulated sugar.
Place the dish uncovered in the oven and bake for 50 to 55 minutes, or until the top crust is deeply golden and the filling is visibly bubbling at the edges.
Once removed from the oven, allow the cobbler to sit undisturbed for at least 15 minutes. This helps the filling set slightly. Serve warm, ideally with a scoop of vanilla ice cream or a dollop of whipped cream.