Preheat oven to 350°F. Butter a 9-inch springform pan. Line with parchment. Butter parchment.

Bring 1/4 cup (1/2 stick) of butter and brown sugar to a boil in a saucepan over medium heat, stirring often. Pour into prepared pan, spreading evenly.

Slice the whole lemons very thinly and discarding seeds; arrange on bottom of the pan, working from the center out, and overlapping lemon slices by approximately 50%. Set aside.

Rub lemon zest into sugar until it is like wet sand.

Beat butter in a stand mixer fitted with the paddle attachment on medium until smooth and creamy. With the mixer running, slowly add sugar and beat until pale and fluffy, about 4 to 5 minutes.

Meanwhile, in a medium bowl, whisk together flour, cornmeal, baking powder, and salt. Add vanilla extract to buttermilk.

Add eggs, one at a time, beating on medium until completely incorporated and smooth, about 1 minute, before adding the next.

Add one-third the flour mixture, beat on low until just incorporated, about 15 seconds, then add half the buttermilk and vanilla, beating on low until just incorporated. Repeat adding flour and buttermilk in the same way, then finish up by beating in the last third of the flour mixture.

Pour and scrape batter into pan, leveling the top.

Place pan on baking sheet and bake, 55 to 60 minutes, or until dark brown on top, and the cake feels firm to the touch and shows no sign of jiggling when moved.

Cool at least 2 hours. Release sides of the springform pan, and invert onto a plate. Remove parchment.