Preheat the oven to 180°C (Electric oven | Electric fan oven 160°C | Gas Mark 4) and place 12 paper cases into a muffin tin.
Using an electric whisk, food processor or wooden spoon, beat the butter, sugar and 1/2 the lemon zest (save 1/2 for the icing) until very light and fluffy.
Add the eggs one at a time, beating each one in well before adding the next.
Carefully fold in the flour (and baking powder if using).
Alternatively, you can mix the ingredients together using a food processor. Use the pulse button to mix together in between adding the ingredients in the order above.
Bake in the oven for 10 - 20 minutes. After 10 minutes, check to see if the lemon cupcakes are ready by inserting a cocktail stick into one of the cupcakes. If it comes out dry then the cupcakes are done. If not, then back in the oven for a few minutes more. Don't overcook the cupcakes otherwise they will dry out. Lift the lemon cupcakes out of the muffin tin and leave to cool on a wire rack.
Make the lemon butter icing
Using an electric whisk, food processor or wooden spoon, beat the butter until very soft.
Gradually add the icing sugar and the rest of the lemon zest and beat into the butter. Add some of the juice of the zested lemon until you have a smooth paste. When the cakes are cool, drizzle some lemon icing mixture over the top. These cupcakes are best eaten on the day they are made.