Preheat the oven to 350 degrees Fahrenheit. Line the base of a 10-inch springform pan with parchment paper and brush the sides with olive oil. Cut the eggplant into roughly ½-inch cubes. Heat the olive oil in a large sauté pan over medium-high heat and sauté the eggplant, stirring occasionally, until it is well caramelized and tender, roughly 10 minutes. If you don’t have a pan large enough to hold all the eggplant, work in batches. If you split your eggplant into batches, return it all to the pan once cooked and add in the sugar, continuing to cook over medium-high heat. Stir frequently and cook until it takes on a dark amber hue, about 5 minutes. Next, add the coffee and continue cooking over medium-high heat until the eggplant is coated in a thick caramel. Spread the mixture evenly across the bottom of the lined springform pan. In a large bowl, whisk together the eggs and both brown and white sugars until smooth. Add the olive oil, buttermilk, and lemon zest, whisking until well incorporated. In a separate bowl, whisk together the flour, salt, baking powder, and baking soda. Add the dry mix to the wet and whisk until just incorporated, then pour the batter over the eggplant in the springform pan. Bake the cake for 55 to 60 minutes. The sides will start to pull away from the pan and the top of the cake will become slightly domed. Once baked, a cake tester will come out clean when inserted into the center, and the surface should spring back when pressed gently with a finger. (Such a moist cake affords a larger window for baking—err on the side of an extra 5 minutes when in doubt.)