Preheat the oven to 180°C (350°F). Line a 22 x 12 cm loaf tin with baking paper.
In a bowl, beat the butter and 225 g caster sugar with an electric mixer until pale and fluffy, then add the lemon zest and vanilla and mix briefly.
Add the eggs one at a time, beating well after each addition and scraping down the bowl as needed.
With the mixer on low, add the yoghurt and the flour and baking powder in 2–3 additions, mixing only until just combined to avoid overmixing.
Transfer the batter to the prepared loaf tin and spread it evenly to the edges.
Bake for 40–45 minutes until golden and firm; a skewer inserted into the center should come out clean. If the top browns too quickly, loosely cover with foil.
While the cake bakes, make the drizzle by mixing 40 g caster sugar with 3 Tbsp lemon juice until the sugar dissolves.
As soon as the cake comes out of the oven, poke holes all over the top with a skewer and brush the hot drizzle over the cake so it soaks in.
Leave the cake in the tin for 10 minutes, then remove and cool completely on a rack.
When cool, make the lemon icing by stirring 150 g sifted icing sugar with 2–3 Tbsp lemon juice to a spreadable consistency, then spread over the cake allowing some to drip down the sides.