Peel and slice fresh peaches, or drain canned peaches well, and set aside. Preheat oven to 350°F (175°C).
In a large bowl, whisk together flour, sugar, baking powder, salt, cinnamon, and nutmeg. Add milk, melted butter, and vanilla extract, stirring until smooth, then gently fold in the peaches.
Grease a 9×13-inch baking dish and pour in the peach batter evenly.
In a small saucepan, combine brown sugar, butter, and boiling water, stirring until melted and smooth. Pour the sauce carefully over the batter without stirring.
Bake 45–50 minutes until the top is golden and firm and the bottom is gooey, checking that edges bubble and a toothpick comes out mostly clean with a few moist crumbs.
Let cool 10–15 minutes, then serve warm, optionally with vanilla ice cream or whipped cream.