Butter a 5 cup capacity ovenproof dish and set aside. Preheat the oven to 180°C.
Place egg whites into the Thermomix bowl and whip 4 min/Butterfly/37°C/speed 3. Remove from bowl and set aside.
Place the butter into the Thermomix bowl and melt 2 min/45°C/speed 1. Insert the Butterfly again.
Zest the limes directly into the Thermomix bowl and add the yolks and sugar. Mix 3 min/Butterfly/speed 3. Remove Butterfly.
Add juice, lime oil, flour and milk and blend 10 sec/speed 4. Return the whipped egg whites to the bowl and combine 17 sec/Interval/dough setting until just combined. It will look lumpy and a little separated. This is ok.
Transfer to the greased baking dish and place into a larger baking dish containing cold water to come halfway up the sides. Bake 40 minutes until golden and just set.
Dust generously with icing sugar and it is best served warm, though no-one complained about it when cold either. Completely delicious.