Preheat the oven to 450 degrees Fahrenheit. In a 10-inch cast-iron skillet, heat 2 tablespoons butter over medium heat until hot, but not brown. Add the berries, ¼ cup sugar and 1 teaspoon vanilla extract. Cook, until the berries are slightly softened (about 2 minutes). Transfer the berries to a bowl.
Wipe out the skillet and place it in the oven.
While the skillet is heating in the oven, using a stand mixer, whisk the eggs until pale and frothy, about 1 minute. Whisk in the milk, flour, salt ¼ teaspoon cinnamon, 1 tablespoon sugar, and remaining ½ teaspoon vanilla until smooth, about 1 minute (the batter will be thin).
Remove the skillet from the oven and add the remaining 4 tablespoons of butter. Swirl to coat the bottom of the pan with the butter (until melted). Pour in the batter, return the skillet to the oven. Bake until puffed and golden-brown, 18 to 20 minutes.
While the Dutch Baby cooks, in a small bowl, stir together the remaining 1 tablespoon sugar and the ¼ teaspoon cinnamon.
Remove the Dutch Baby from the oven, top with the berries and sprinkle with the cinnamon-sugar mixture. Serve immediately.