Preheat the oven to 350 degrees F. Butter and flour a 9-inch spring form pan. For this recipe you will need the juice and zest from two bergamot oranges.
Place the sugar in a large bowl. Zest the bergamot orange onto the sugar and use your fingers to rub the zest into the sugar. Add the flour, salt, baking powder, and baking soda and whisk for 30 seconds to thoroughly combine the ingredients.
Place the eggs in a large bowl and whisk. Add the olive oil, the milk and the orange juice and whisk the ingredients until the they are well combined. Add the dry ingredients and whisk together until no bits of flour remain, but do not overmix or you will end up with a tough cake.
Pour the batter into the prepared pan. Use an offset spatula to smooth the top. Bake the cake for 45-50 minutes until a cake skewer inserted in the center comes out clean, with just a couple of crumbs clinging to it.
Ten minutes before the cake is done combine the bergamot orange juice, the sugar and the butter in small saucepan and place over medium heat. Heat the mixture until the butter is melted the sugar is dissolved. Do not boil the mixture.
Remove the cake from the oven and let it cool the cake in the pan for 10 minutes. Run a spatula around the edges of the cake to loosen it from the sides of the pan and remove the ring from the cake. Brush the cake with the syrup and allow it to completely before frosting.
At this point, you can enjoy the cake unfrosted. If making the cake in advance it can be stored at room temperature wrapped in plastic wrap for two days.