Heat the oven to 350°F (177°C) and lightly butter a 10 to 12 inch oven-safe skillet. A 10 inch springform pan also works, but a 9 inch pan will be too small.

In a medium bowl, whisk the flour, cornmeal, baking powder, baking soda, and salt together. Set aside.

Using a hand mixer or a stand mixer with the paddle attachment, beat the butter and brown sugar on high speed until light and creamy, about 2 minutes, scraping down the bowl as needed. Add the eggs and vanilla, then beat on medium-high until fully combined. Reduce the mixer to low, add the dry ingredients until just combined, and then slowly pour in the buttermilk. Stop the mixer as soon as the batter is smooth; do not overmix. Fold in the chopped peach and blackberries by hand. Spread the thick batter into the prepared skillet and sprinkle with coarse sugar and a few extra blackberries, if desired.

Bake for 30 to 35 minutes, or until a toothpick inserted in the center comes out clean. A thinner cake in a 12 inch skillet will often be done closer to 30 minutes.

Set the skillet on a wire rack and let the cake cool slightly before icing and serving.

Cut the icing butter into small pieces and melt it in a light colored skillet over medium heat, stirring occasionally. Once the butter foams and then develops brown specks at the bottom with a nutty aroma, about 5 to 6 minutes, remove it from the heat immediately and cool for 5 minutes. Whisk in the confectioners' sugar, milk, vanilla, and salt until smooth. If needed, add a little more sugar to thicken the icing or more milk to thin it out.

Drizzle the brown butter icing over the warm cornbread, then slice and serve warm or at room temperature. Cover any leftover cake tightly and refrigerate for up to 5 days.