Preheat the oven to 425° and heat a 10-inch cast-iron skillet over moderate heat. In a medium bowl, whisk the eggs with the finely grated lemon zest, sugar and salt until combined. Add the flour and milk and whisk until smooth. Add 1 cup each of the raspberries and blackberries.
Melt the butter in the skillet and add the batter, spreading the fruit evenly. Bake in the center of the oven for about 22 minutes, until the edges are browned and puffed and the center is lightly browned in spots. Transfer the skillet to a trivet and dust the Dutch Baby with confectioners' sugar. Cut into wedges and serve with the remaining fresh berries.