Preheat your oven to 170°C (150°C fan).
Grease and line an 18 cm deep, loose-bottom or springform cake tin with baking parchment.
In a bowl, cream the butter and caster sugar with an electric mixer until light and fluffy, around 4 minutes.
Add the eggs, one at a time, making sure to beat thoroughly after each addition. Stir in the lemon zest and vanilla.
Gently fold in the flour, baking powder, salt, and Greek yogurt, mixing just until the ingredients are incorporated.
Transfer the batter to the prepared tin and smooth the surface with a spatula.
Bake in the center of the oven for 50-55 minutes, or until the top is golden brown and the cake feels firm to the touch.
Allow the cake to cool in the tin for 10 minutes, then transfer it to a wire rack to cool completely.
Sift 400g of icing sugar into a bowl and gradually mix in lemon juice until you reach a thick, pourable consistency.
Place the cake on a stand or plate, spoon the icing over, and let it drip down the sides.