Heat the oven to 400°F (200°C). Set the butter in a 6.5-inch cast iron skillet or 5×5-inch baking dish and slide it into the oven to melt the butter and heat the pan.
In a small bowl, whisk together the egg, milk, vanilla, and sugar. Add the flour and salt and whisk until the batter is smooth.
Carefully take the hot pan out of the oven and swirl the melted butter over the bottom and up the sides. Scatter the chopped peach evenly across the bottom.
Pour the batter over the peach and return the pan to the oven. Bake for about 20 minutes, until the edges are puffed and golden and the center is set.
Lift the Dutch baby onto a plate or cooling rack with a spatula and let it rest for about 5 minutes before serving.
Notes
Get the pan hot first. Heat the skillet or baking dish with the butter in it before the batter goes in. A hot pan is what makes the Dutch baby climb the sides and puff.Use a small pan. Stick with a 6.5-inch cast iron skillet or a 5×5-inch oven-safe baking dish. A larger pan spreads the batter too thin and you lose the puff.Whisk the batter smooth. Whisk until the batter is thin and pourable with no flour lumps, which bake into dense spots.Reach for a ripe peach. A ripe peach is softer and sweeter, so it cooks down into the pancake and you need less added sugar.Don't overload the pan with fruit. One peach is enough. Too much fruit adds weight and moisture that keep the Dutch baby from puffing, so if you use canned or frozen peaches, pat them dry first.Watch for puffed, golden edges. The Dutch baby is done when the edges are puffed and golden, usually about 20 minutes. Go by how it looks rather than the clock.If doubling this recipe, use a 10-inch cast iron skillet.