Preheat the oven to 180°C (356°F) on top and bottom heat.
Whisk the softened butter, sugar, and vanilla in a medium-sized bowl until light and fluffy (I use a hand whisk).
Add one egg at a time, incorporating each in between.
Top the batter with the juice and zest from 1 lemon, along with the Greek yogurt. Whisk again.
Fold in the flour and baking powder.
Pour the batter into a prepared baking tin lined with parchment paper.
Bake for 35-38 minutes, until the center is set and a toothpick comes out clean.
Syrup
Add all the ingredients to a small pot over low-medium heat until the sugar dissolves (about 2 minutes).
Remove from the heat and let it cool slightly.
Icing
Mix the lemon juice with powdered sugar to make a thick, white icing. You may need to add a few more drops of lemon juice or water to make it just right (depending on the size of your lemon).
Assemble
Stick a skewer into the top of the cake (about halfway through). I made about 50 small holes.
Spoon the syrup over the warm cake. Let the cake cool completely.
Once cool, pour the icing on top of the cake. You can let it drip over the sides or leave it only on the top.