For the cake: Butter and flour an 8-inch round cake pan.
In a medium bowl, whisk together the cornmeal, cake flour, minced rosemary, baking powder, and salt.
Using a stand mixer with a paddle attachment, on low speed beat the butter and vanilla together until combined. Slowly add the powdered sugar. Once the sugar is incorporated increase the speed to high and beat until fluffy, about 3 minutes. Add the egg yolks and eggs, 1 at a time. Reduce the speed to medium and add the sour cream. On low speed add the dry ingredients just until incorporated.
Pour the batter into the prepared cake pan and smooth the surface with a spatula. Bake in the lower third of the oven until the cake is golden and pulls away from the sides of the pan, about 35 minutes. Transfer the pan to a wire rack and let cool. Transfer the cake from the pan to a serving plate and dust with powdered sugar.
For the Balsamic Syrup: Place the sugar, balsamic vinegar, and rosemary in a small saucepan. Bring to a boil and simmer until the sugar is dissolved, about 5 minutes. Discard the rosemary sprig and let the syrup cool.
To serve, place a slice of cake on a serving plate; spoon some syrup around the plate.