In a 10-inch well-seasoned cast-iron or non-stick skillet, melt the butter over medium-high heat, add brown sugar and stir until sugar dissolves. Add the balsamic and cherries, bring just to a boil. Remove from heat, let cool.
Meanwhile in a medium bowl, mix together the flour, cornmeal, baking powder, and salt. In a large bowl, using an electric mixer beat the eggs, egg yolk, milk, olive oil, and vanilla extract until frothy. Add sugar and mix to combine. Slowly pour in dry ingredients, mix until smooth. Pour batter into cooled skillet over cherries and place in the middle of the oven to bake until a toothpick inserted into the center of the cake comes out clean, about 30 minutes.
Remove skillet from the oven and allow to cool on a wire rack for 20 minutes. Place a plate over the top of the skillet and, working quickly, invert the plate and skillet. If the cake does not come out easily, give the skillet a quick tap on a solid surface to help dislodge the cake. Allow to cool to warm before serving. Serve warm or at room temperature.