Preheat oven to 400º F and lightly grease a 9×13-inch baking dish with butter or non-stick spray.
In a large saucepan over medium heat, whisk together brown sugar, sugar and nutmeg, then mix in butter and water and bring to a boil, stirring until sugar is fully dissolved.
Remove from heat and set aside.
In a large bowl, whisk together flour, sugar, baking powder, salt, cinnamon and nutmeg until combined.
Whisk in milk and vanilla extract until smooth, then fold in diced peaches.
Pour filling into greased baking dish, then pour topping on top.
Place in oven and bake for 50-55 minutes, covering with aluminum foil, if necessary, until cooked through.
Remove from oven and let cool 15 minutes before serving.