Preheat oven to 180ºC (356ºF). Line a round 9-inch baking pan with parchment paper, and set aside. NOTE: Alternatively, if you have a non-stick pan, liberally grease it with melted butter and dust it with a light layer of flour. This will leave your finished cake with smooth sides.
In a large mixing bowl, add sugar, eggs, lemon zest, and lemon juice.
Using electric hand beaters, beat on low-medium speed for 30 seconds until well combined. Scrape down the sides if needed with a spatula.
With the beater running on low-medium speed, slowly drizzle in the olive oil in a steady stream to emulsify, about 30 seconds in total.
Add yogurt and vanilla extract, and beat for 15 seconds until well combined. Scrape down the sides if needed.
Sift over flour and baking powder. Add the salt.
Beat until incorporated, about 15 seconds but do not over-mix, just until it’s incorporated into a smooth batter and no lumps of flour remain. Scrape down the sides.
Pour batter into the prepared baking pan. Smoothing out the top.
Bake for 30-35 minutes, until a toothpick inserted into the middle, comes out clean and it’s lightly golden on top.
Remove from the oven, set aside and immediately make the lemon syrup.