Puree the banana and ¼ cup of the milk in a food processor until smooth. Combine the banana puree, the remaining 1¾ cups of milk, ½ cup of the brown sugar, the cinnamon, cardamom, nutmeg, and salt in a medium, heavy saucepan. Scald the milk mixture over medium-high heat, stirring often, for 5 minutes.
While the milk is scalding, whisk together the egg yolks in a medium mixing bowl. Add the remaining ½ cup of brown sugar and whisk until the eggs are light and fluffy. Whisking constantly, add a small amount of the hot milk to the egg mixture. Gradually whisk in the remaining hot milk.
Return the custard mixture to the pan and cook over medium-low heat, stirring often, until the custard is thick enough to coat the back of a spoon, about 5 minutes. Remove the pan from the heat and place in a bowl of ice water to quickly cool the custard. Let the custard cool, stirring often, for 5 minutes. While the custard is cooling, combine the cream and the vanilla in a medium bowl. Stir in the custard mixture. Cover with plastic wrap and press the wrap directly onto the surface of the custard. Refrigerate until completely chilled, at least 4 hours or overnight. The custard may be stored in the refrigerator for up to 3 days.
Freeze the chilled custard mixture in an ice-cream maker according to the manufacturer’s instructions. Transfer the ice cream to an airtight container and freeze in the freezer for 2 to 4 hours before serving.
Recipe from Spice Dreams by Sara Engram and Katie Luber/ Andrews McMeel Publishing, 2010.