How to Make It
Preheat the oven to 375°. Butter and flour a 9-inch springform pan. Cut the top and bottom off the orange and discard. Thinly slice the orange and discard any seeds. Transfer the orange slices to a food processor and puree. Add 1/2 cup of the sugar and the olive oil and pulse just until combined.
In a medium bowl, whisk the flour with the baking powder. In another medium bowl, using a handheld electric mixer, beat the eggs with the remaining 1 cup of sugar at moderate speed until thick, about 8 minutes. Fold in half of the flour mixture, then fold in the orange puree and vanilla. Fold in the remaining flour mixture.
Scrape the batter into the prepared pan and bake for 20 minutes. Lower the oven temperature to 325° and bake the cake for 30 minutes longer, or until golden brown on top and a toothpick inserted in the center comes out clean. Transfer the cake to a rack to cool, then unmold the cake and serve.