Pour cream and half and half into heavy medium saucepan. Bring mixture just to simmer. Whisk yolks and sugar in large bowl to blend. Gradually whisk in hot cream mixture. Return mixture to same saucepan and stir over medium-low heat until mixture thickens and leaves path on back of spoon when finger is drawn across, about 7 minutes (do not boil). Strain custard into clean large bowl. Stir in rum and vanilla. Refrigerate until cold, about 2 hours. Puree bananas and lemon juice in processor until smooth. Stir into custard. Process in ice cream maker according to the manufacturer's instructions. Transfer to covered container and freeze. (Can be prepared 3 days ahead.) Makes about 2 quarts.