Preheat a fan-forced oven to 160C (320F) and line a 20cm square cake tin with parchment paper.
In a stand mixer, whisk eggs and caster sugar on high speed for 8 to 10 minutes until thick, pale, and the mixture holds a ribbon trail.
Gently fold in the triple-sifted flour and lime zest using a large metal spoon, taking care not to deflate the aeration.
Drizzle the melted butter around the edge of the bowl and fold in until just incorporated.
Transfer batter to the prepared tin and bake for 22-25 minutes until the sponge springs back when lightly pressed.
Allow the sponge to cool in the tin for 5 minutes before transferring to a wire rack to cool completely; for best results, refrigerate for 2 hours to firm up before cutting.
Slice the sponge into 16 even squares and then halve each square horizontally.
Spread a thin layer of strawberry jam on the bottom half of each square and replace the top half to form a sandwich.
To make the glaze, whisk together the icing sugar, strawberry puree, lime juice, and boiling water in a heat-proof bowl over a pot of simmering water until smooth and slightly fluid.
Place the desiccated coconut in a wide, shallow bowl.
Using a fork or dipping tool, submerge each cake square into the strawberry-lime glaze, drain the excess, and immediately roll in the desiccated coconut until fully coated.
Place the finished lamingtons on a wire rack for 30 minutes to allow the glaze to set completely.