Preheat the oven to 350F/180C.Butter a 9x5 inch loaf pan and line it with parchment paper, leaving some excess parchment paper hanging over the sides so that you can lift the cake out when it's done. You can use either a 9x5 inch loaf pan or an 8x4 inch loaf pan, which will give you a taller cake.
Using an electric beater, cream the butter and sugar in a large mixing bowl. Add the lemon zest and vanilla extract and beat until combined. Add the eggs one at a time and beat until combined. Add the oil and sour cream and beat until combined.In a separate bowl, combine the flour, baking powder and salt.Add half of the flour and the milk, alternating between the two, and beating after each addition just until combined. Be careful not to over-beat the batter or the cake will be dense.
Spoon the cake batter into the loaf pan and smooth the batter.Bake the cake on the middle shelf for 45-55 minutes or until a toothpick inserted into the center of the cake comes out clean. Do not over-bake or the cake will be dry. While the cake is baking, prepare the lemon syrup by stirring together the lemon juice and caster sugar until the sugar is dissolved.Take the cake out of the oven and while it's still hot, use a skewer or toothpick to poke multiple deep holes throughout the entire cake.While the cake is still hot, brush the lemon syrup all over the top of the cake, letting it soak into the cake. Use all of the syrup.Let the cake sit until it is fully cooled.
Once the cake is fully cool, prepare the lemon icing by stirring together the powdered sugar and lemon juice until it is smooth. This can also be done in advance.Grab the parchment paper on each side and lift the cooled cake out of the loaf pan and transfer it to a wire rack or work surface.Drizzle the lemon icing all over the cake and use a knife or icing spatula to smooth it all over the top and sides of the cake.Sprinkle some extra lemon zest over the top of the cake while the icing is still wet (optional but adds more flavor and a nice aesthetic). Let the cake sit until the icing is set.Slice and serve.Keep the cake stored in a sealed container at room temperature to prevent it from drying out. It will keep for up to 4 days.
Notes
Freezing: This cake can be frozen for up to 3 months, but should be done prior to adding the final lemon icing. Freeze the cake, let it thaw overnight, and then ice it.