In a heavy-bottomed saucepan, combine cream, milk and sugar. Scrape seeds of vanilla bean into cream mixture and drop in the seedpod also. Place saucepan over medium-high heat. As soon as mixture begins to boil, remove pan from heat.
Beat egg yolks in a small bowl and slowly, in a very thin stream, add 1 cup of the hot cream mixture, stirring constantly to prevent the cream from scrambling the eggs. Gradually pour yolk and cream mixture back into saucepan, stirring constantly. Cook over medium heat for 8-10 minutes, stirring constantly, until mixture begins to thicken slightly, coats the back of a spoon and reaches 210 F on an instant-read thermometer.
Remove pan from heat. Pour cream mixture through a fine mesh strainer into a large bowl and let cool for 10-20 minutes. Cover and chill in refrigerator for 2-3 hours.
Follow the manufacturer's instructions for your ice-cream maker. Pour in the chilled cream mixture and let the machine churn for 20-25 minutes. Add the banana and chocolate when the ice cream is 3/4 frozen. Let the machine churn for another 5-10 minutes maximum. Cover the surface of the ice cream with wax paper or plastic wrap before storing in freezer to prevent ice crystals from forming.