In a bowl whisk together the sugar, the cornstarch, the yolks, and the whole eggs and add the milk, scalded, in a stream, whisking. In a heavy saucepan bring the mixture to a boil over moderate heat, whisking, and boil it, whisking, for 1 minute. Add the rum, boil the custard for 1 minute more, and strain it through a fine sieve into a metal bowl set in a larger bowl of ice and cold water. Chill the custard until it is cold and stir in the vanilla and the cream. In a food processor puree the bananas, stir the puree into the custard , and chill the mixture until it is cold. Freeze the mixture in an ice cream freezer according to the manufacturer's instructions.

Makes about 1 1/2 quarts.