Prepare a 9-inch cake pan: brush all around with oil and line the bottom with parchment paper—oil the paper too.
Preheat oven to 350ºF. Measure the flour, cornmeal, sugar, baking powder and salt into a large bowl. Stir with a fork, then set aside.
Heat the butter in a small pan until it is just melted. Mix in the six tablespoons of olive oil.
Break the eggs into a medium bowl and beat them with the lemon juice, the lemon zest and the extracts until everything is uniform, about a minute or two. Mix in the butter and oil. Stir in the buttermilk.
Add in the dry ingredients in two or three batches and stir together quickly, just until everything is mixed together. Gently stir in the blueberries.
Scrape the batter into the prepared cake pan. Put in the oven and set a timer for 30 minutes. It make take longer! The cake is ready when the top is lightly golden, the cake is just barely pulling away from the sides, and a cake tester inserted near the center comes out clean.
While the cake is baking, make the syrup. Put the water and sugar in a small saucepan, drop in the rosemary, bring to a simmer and cook over low heat for 10 minutes. Remove from heat, strain into a small bowl, discard the rosemary and set the syrup aside to let it cool.
When the cake is done, let it rest in the cake pan on a rack for 15 or 20 minutes. Run a thin knife around the edge to loosen the sides, then remove from the pan and set onto a serving plate.
Poke the cake all over with a toothpick or the cake tester—you want to make a lot of little holes all over. Drizzle rosemary syrup on top (we used about half the syrup), letting it slowly soak in. You can use it all or you can skip this entirely. We liked the little extra sweetness and the little subtle extra infusion of rosemary. The original inspiration for this cake, in the Hedgebrook cookbook, suggests passing the extra syrup when you serve the cake, but we think it's already sweet enough.