Heat oven to 180°C (350°F) with bottom heat if possible, otherwise both top and bottom heat. Lightly grease the baking pan, line it with baking paper, and let the longer sides overhang.
In a bowl, beat the soft butter and sugar with a mixer for 2 to 3 minutes until light and foamy. Add the egg and mix just until combined. Add the flour and desiccated coconut and mix with a wooden spoon just until combined.
Pour the batter into the prepared pan and smooth the surface with the back of a spoon. Bake for 15 minutes until golden brown.
While the base bakes, make the lemon curd: combine butter, sugar, egg, lemon zest, and lemon juice in a saucepan. Heat over medium heat, stirring constantly, until boiling. Simmer for 5 minutes, stirring, until thickened.
Remove the lemon curd from heat and let cool for 5 minutes. Remove the cake pan from the oven and spread the lemon curd evenly over the baked base. Keep the oven at 180°C (350°F).
In a bowl, beat egg and sugar together with a fork. Stir in the desiccated coconut and spread this mixture evenly over the lemon curd.
Return the pan to the oven and bake for 20 to 25 minutes until the topping is golden brown and firm.
Remove from the oven and let cool in the pan for 45 minutes until completely cool. Cut into squares in the pan. Use the overhanging baking paper to lift the entire cake out of the pan.