A polenta cake from pastry chef Hannah Buoye, of A16 Rockridge in Oakland, was the model for this dense, moist dessert. As the sliced lemons bake under the cake and absorb butter and brown sugar, they take on a marmalade-like quality. Be sure the cake is fully baked before you remove it from the oven (see step 6), or it may sink.
1. Preheat oven to 350°. Spray inside of a 9-in. springform pan with cooking-oil spray. Snugly line pan with a 12-in. circle of parchment paper*, pressing pleats flat. Spray parchment with oil; set aside.
2. Bring 1/4 cup butter and the brown sugar to a boil in a small saucepan over medium heat, stirring constantly. Pour into lined pan and spread evenly.
3. Thinly slice 2 lemons crosswise, using a handheld slicer and removing seeds with a knife tip as you go. Discard ends. Set a small lemon slice in center of butter mixture in pan. Arrange more slices in overlapping circles to fill pan (each slice should overlap the previous one by half). Save any extra lemon for other uses.
4. Beat remaining 1/2 cup butter, the granulated sugar, and lemon zest in a large bowl with a mixer on medium speed to blend, then on high until pale and fluffy, 3 to 4 minutes. Add 1 egg at a time, scraping inside of bowl and beating well on high speed, about 1 minute per egg. In a medium bowl, whisk together flour, cornmeal, baking powder, and salt. Combine milk and vanilla.
5. Add a third of flour mixture to butter mixture and blend on low speed; scrape inside of bowl. Blend in half of milk mixture. Repeat to incorporate remaining ingredients, ending with flour mixture. Pour batter into pan; spread evenly.
6. Bake until cake springs back in center when lightly pressed, 50 to 55 minutes (it will be well browned). Let cool in pan on a rack at least 2 hours. Run a thin knife between parchment and pan; release rim. Invert cake onto a plate. Remove parchment and cut cake with a serrated knife.
*For an even circle, use a pencil to trace around a 12-in. plate onto parchment.
Make ahead: Up to 1 day.