If using an electric ice cream maker, place the body into the freezer overnight. It needs to be completely frozen to freeze the ice cream.
Combine the cream, milk, and 1 cup of the sugar in a medium, heavy bottom saucepan. Scrape the seeds of the vanilla bean into the cream mix and add in the bean itself. Bring to a light boil over medium high heat, watching carefully so the mixture doesn't boil over. Remove from the heat.
Beat the yolks in a mixing bowl and slowly whisk in 1 cup of the hot cream mixture. Gradually add the egg mixture back into the pot of cream. Cook over medium heat, stirring occasionally, until the mixture coats the back of a spoon or reaches 170 degrees F, about 6 to 8 minutes. Remove from the heat and place in a container. Cover and place in the refrigerator to cool for about 2 to 3 hours.
Put the ice cream maker together and turn it on. Pour in the cream mixture and let churn for about 20 to 25 minutes, according to manufacturer's instructions. Stir in the banana and chocolate when the ice cream is 3/4 of the way frozen.