Mix 3 tablespoons of coconut milk with cornflour to form a smooth paste.
Heat remaining coconut milk, coconut cream, and paste gently in a saucepan.
Whisk egg yolks, agave nectar, and caster sugar in a bowl.
Slowly whisk warmed coconut milk into the egg mixture until combined.
Stir mixture over iced water until slightly thickened and cooled.
Blend mango purée, lime juice, zest, and cooled milk mixture until smooth.
Churn in an ice cream maker according to manufacturer instructions.
Roll ice cream in cling film into a 30cm long cylinder and freeze.
Boil pineapple, juice, jam sugar, and star anise for 20 minutes until translucent.
Stir in vanilla and lime zest, cool, then blend until smooth.
Whisk egg whites with sugar to stiff peaks then fold in flour and yellow coloring.
Pipe diagonal lines on a lined 33x23cm tin and freeze for 30 minutes.
Preheat oven to 200°C.
Whisk egg whites and sugar to stiff glossy peaks for the joconde.
Whisk almonds, icing sugar, and whole eggs till doubled, then gently fold in flour.
Fold in the reserved egg whites and spread over the frozen pattern.
Bake for 8 to 10 minutes until golden.
Turn out onto a rack, peel off parchment, and cool completely.
Spread pineapple jam over the sponge and roll around the unwrapped ice cream cylinder.
Freeze the completed roll until hardened and serve in slices.