In a saucepan, combine the cream and milk and warm gently over a low flame.
Place the sugar and egg yolks in a mixing bowl and beat until smooth in texture and pale yellow in color. Slowly add about 1 cup of the cream and milk mixture to the egg mixture and whisk vigorously to combine.
Pour the egg mixture into the saucepan with the remainder of the cream and milk mixture and whisk continuously until the custard starts to thicken and can coat the back of a spoon, about 3 minutes. Add the chocolate chips, banana puree, and the banana extract to the custard and stir to combine. Cook for 1 minute longer, then remove from the heat, and strain through a fine mesh sieve.
Place the custard in a non-reactive container, cover with plastic wrap and chill in the refrigerator for at least 2 hours. Remove from the refrigerator and mix in an ice cream machine according to the manufacturer's instructions. Once frozen, place the ice cream in a plastic tub to continue freezing in the freezer.
Yield: 1 quart