adapted from a recipe on saveur.com
2 1/3 cups sugar
unsalted butter, for greasing
1 1/2 cups flour, plus extra for the pan
2 teaspoons baking powder
1 teaspoon baking soda
1 teaspoon vanilla extract
1/4 cup fresh orange juice
1/4 cup powdered sugar
Trim about half an inch from the tops and bottoms of oranges and cut them into quarters, lengthwise. Put into a 4-quart saucepan and fill with 6 cups of water. Bring to a rapid boil. Drain the water and repeat this process a total of 3 times. After the last time, add 1 cup of sugar and 4 cups of fresh water into the pan. Cook over medium high heat, stirring occasionally, until sugar dissolves and orange rinds can be easily pierced with a fork; about 30 minutes. Remove pan from heat and let cool to room temperature. You can also do this a day ahead and keep the oranges with syrup in the refrigerator overnight.
Heat oven to 350°F. Grease a 10-inch round cake pan with butter, dust with flour and line the bottom of the pan with a piece of parchment paper cut to fit. Set pan aside.
Whisk together flour, baking powder and baking soda in a a medium bowl and set aside. Remove the orange segments from the syrup, removing any seeds, but leaving the peels intact. Put the orange pieces into a food processor and pulse until the oranges form a chunky puree. Add remaining sugar, reserved flour mixture, vanilla and eggs and process in the food processor until incorporated, about 3-4 minutes. Add the olive oil and process until thoroughly combined. Pour batter into your prepared pan and bake until a toothpick inserted in the center comes out clean; about 40-45 minutes.
Let cake cool for 30 minutes while you whisk the fresh orange juice and powdered sugar into a thin glaze. Transfer the cake to a plate and using a pastry brush, brush the glaze over the top and sides of cake. Let the cake cool completely before serving.