Preheat an oven to 325 degrees F (165 degrees C). Put the butter in an 8x8 inch baking dish and place in the oven to melt while you prepare the remaining ingredients.
Bring the peaches with syrup and 2 tablespoons of sugar to a boil in a saucepan. Reduce heat to medium-low and simmer 10 minutes; set aside. Whisk together the flour, 1 cup sugar, cinnamon, and nutmeg in a bowl. Stir in the milk until the mixture is just moistened. Pour into the hot baking dish over the butter. Spoon the peach slices and most of the syrup over the batter.
Bake in the preheated oven for 30 minutes. Sprinkle with the remaining 2 tablespoons of sugar. Return to the oven and bake until the top is golden brown, 30 to 45 minutes longer.