Preheat the oven to 325°F.
In a large bowl, combine sugar and lemon zest. Using a wooden spoon or rubber spatula, press the lemon zest into the sugar. This will release the oils into the sugar and become fragrant.
Whisk in the eggs until combined.
To the same bowl, add in the rest of the wet ingredients; vanilla, half cup of lemon juice, good quality olive oil, yogurt and mix on medium-high speed until well combined.
Finally add in the dry ingredients; flour, baking powder and salt. Mix again until fully incorporated.
Pour batter into a prepared 9 inch springform pan or cake pan.
Bake for 60 minutes until golden brown, or until a toothpick comes out clean after testing the center of the cake.
Allow cool on a wire rack before removing from the pan and allow to cool completely before adding any icing or glaze.