In a large mixing bowl, combine the sliced peaches, granulated sugar, brown sugar, vanilla extract, ground cinnamon, ground nutmeg, lemon juice, and cornstarch. Toss gently to coat the peaches evenly.
Transfer the peach mixture to a 9x13-inch baking dish and spread it out evenly.
In a separate mixing bowl, whisk together the flour, granulated sugar, baking powder, and salt.
Gradually pour in the milk while whisking until the batter is smooth and well combined.
Pour the melted butter into the batter and stir until fully incorporated.
Pour the batter evenly over the peaches in the baking dish. Do not stir; the batter will rise up over the peaches as it bakes.
Bake in the preheated oven for 40-45 minutes, or until the top is golden brown and a toothpick inserted into the topping comes out clean.
Allow the cobbler to cool for at least 15 minutes before serving.
Serve warm with a scoop of vanilla ice cream or a dollop of whipped cream, if desired.
Notes
For best results, use ripe, fresh peaches. Adjust sugar based on the sweetness of the peaches. You can substitute frozen or canned peaches if fresh ones are unavailable.