Start by making the ice cream. Put the milk, cream and star anise into a saucepan and heat until boiling point is reached. Then remove from the heat and infuse for 20 minutes.
Meanwhile, beat the egg yolks and syrup together, then strain in the infused milk and return to the pan. Heat gently, stirring until the custard thickens to coat the back of a wooden spoon. Do not boil. Leave the custard until cold and then freeze in an ice cream maker, according to the manufacturer's instructions.
Place the whole bananas in their skins on the barbecue and cook for 4—5 minutes, turning halfway through until the skins are blackened. Peel and serve the banana with a scoop of ice cream.