Preheat the oven to 175 C. Grease a 9" springform pan very well (or use a regular round pan but springform makes removal much easier and is recommended).
Sift together the flour, baking powder and salt in a bowl. Set aside.
In the bowl of a stand mixer fitted with a whisk attachment, beat the eggs for 1 minute on medium speed till frothy. Add the sugar, lemon zest and vanilla. Beat again for about 3 minutes on high speed until the mixture has thickened and looks very fluffy and pale yellow. If you don't have a stand mixer, you can also use a large, deep mixing bowl with a hand mixer.
Pour in the olive oil with the mixer running on medium and beat till the oil is fully incorporated, about a minute.
Add half the flour mixture and beat on low for 30 seconds to a minute to incorporate, then pour in all of the milk and beat for another 30 seconds.
Add the remaining flour mixture and beat for another minute till you have a smooth, velvety batter. Scrape down the bowl as needed to make sure no flour is stuck to the base. The batter is of a pourable consistency.
Transfer to the prepared pan and sprinkle the top evenly with the extra 2 tablespoons sugar.
Bake for 45 to 50 minutes until the top is golden-brown and flaky (it will rise and crack quite a bit) and a toothpick inserted in the center comes out with a few moist crumbs but no wet batter. If the top is browning too fast, cover loosely with foil and continue baking till the cake is done. You are likely to see some larger patches of sugar that didn't completely melt down, that's ok :)
Allow the cake to cool for 20 minutes (the crust will fall a little), then carefully remove the sides of the pan and allow to cool completely before slicing and serving. Tastes best at room temperature.
Store for 3 to 4 days at room temperature in an airtight tin (the crust is at its crackly best on day one, it will soften after that), but in warm climates, I'd recommend keeping it in the fridge after the first couple of days, where it will last for about 10 days. Bring individual pieces to room temperature before eating. Happy baking!