Preheat an oven to 350 degrees F.
Grease and line the base of an 8-inch deep square baking tin with parchment paper.
In a large mixing bowl, whip the butter and sugar with an electric beater until the mixture is pale and creamy, about 5 minutes. Beat in the lemon zest and vanilla. Add the eggs, 1 at a time, beating until just combined after each addition. Fold in the flour in 2 batches until well combined. Spoon the batter into the tin and bake for 40 to 50 minutes, or until a skewer poked into the middle of the cake comes out clean. Cover the tin loosely with foil if the crust browns too quickly.
Cool for 10 minutes before removing the cake from the tin and turning it out onto a wire rack to cool completely. Ice the pound cake with the lemon butter icing.
Beat the butter with an electric beater until very soft and white. Beat in the sugar, lemon zest, and juice. Spread over the cake.