In a medium mixing bowl, gently toss together the sliced peaches, raspberries, sugar, cinnamon, nutmeg, and lemon juice until the fruit is evenly coated.

Let the peach and raspberry mixture sit at room temperature while you prepare the batter.

Preheat the oven to 350°F. Pour the melted butter into a 9×13-inch baking dish, tilting the pan to coat the bottom evenly.

In a large mixing bowl, whisk together the flour, sugar, baking powder, and salt.

Add the milk and vanilla to the dry ingredients. Stir just until smooth and combined; do not overmix.

Pour the batter evenly over the melted butter in the baking dish. Do not stir.

Spoon the fruit mixture evenly over the batter, making sure to include any juices from the bowl.

Bake for 45 to 50 minutes, or until the top is golden brown, the edges are bubbling, and a toothpick inserted in the center comes out clean.

Let the cobbler cool for at least 15 minutes before serving to allow it to set.

Serve warm with vanilla ice cream or whipped cream, if desired.