Whisk eggs and sugar in a large microwave safe bowl, until well combined. Add juice and butter and microwave uncovered, on medium power (50%) for 6-8 minutes, whisking every minute, until thick. Refrigerate in an airtight container.
Using a stand mixer or electric mixer, beat butter, caster sugar and lime zest until creamy. Add eggs one at a time, beating well after each addition. Add sour cream and beat until combined before folding in flour.
Mix together icing sugar and enough boiling water to form a runny paste. Carefully roll each lamington in icing paste and then in coconut, gently pressing down to coat all over.