Melt butter in a cast-iron skillet in the preheating oven.
Combine peach slices, brown sugar, and cinnamon in a bowl; gently toss to coat the peaches well.
Beat eggs, milk, flour, vanilla extract, salt, and nutmeg together in a bowl until batter well-combined but a little lumpy; pour into skillet. Top batter with the peach mixture.
Bake in preheated oven until set in the middle, about 20 minutes.