Put the mixer on low and add the eggs, one at a time. Beat in each egg for one minute before adding the next one and scrape down the bowl between each addition.
Add the powdered sugar and cooled strawberry reduction to the butter in turns. Continue beating for 3 minutes after it is all mixed in, until the icing is light and fluffy.
Place the next layer on top of the first and proceed to ice the cake with the remaining icing.
Note: This cake can be made over the course of two days. Bake the cake and make the strawberry puree and reduction on the first day. Finish it the next day.