Prep. Add butter to a 9x13 baking dish and place into the oven as it preheats to 375º F.
Sweetened fresh peaches. Meanwhile, add 1/2 cup of the sugar to the peaches in a medium bowl and allow to sit while mixing the peach cobbler crust batter.
Whisk together batter. Whisk together the remaining sugar, the self-rising flour, the milk, and the vanilla. Pour this batter on top of the melted butter and then top with the peaches and their juice. Do not stir.
Bake. Bake until the cobbler is golden brown, about 35 minutes. Serve warm.
If fresh peaches are not available, use 2 (16-ounce) cans of peaches, along with their juice. If using frozen peaches, allow to thaw completely and proceed with recipe directions.