Preheat your oven to 350°F and pour the melted butter into a 9x13-inch baking dish or cast-iron skillet.
In a medium bowl, whisk together the flour, 3/4 cup sugar, baking powder, and salt, then stir in the milk and vanilla extract until just combined to form a batter.
Pour the batter evenly over the melted butter in the baking dish without stirring, then arrange the peach slices on top of the batter.
Sprinkle the remaining 1/4 cup sugar, cinnamon, and nutmeg over the peaches.
Bake for 40-45 minutes until the top is golden brown and the cobbler is bubbling around the edges.
Let cool for 10 minutes, then serve warm with generous scoops of vanilla ice cream.