Bring milk, bananas and brown sugar to a simmer in a large saucepan over medium heat. Cover, reduce heat to low, and simmer 10 minutes.

Let cool 5 minutes, then puree in a blender or food processor until smooth. Return mixture to the saucepan; set over very low heat to keep warm.

Beat egg yolks and sugar in a large bowl with a mixer at high speed until thick.

Gradually beat half the milk mixture into the yolk mixture. Return mixture to the saucepan. Set over low heat and cook until thick and bubbling, about 2 minutes, stirring constantly. Stir in vanilla, salt, and cream.

Pour into a large container. Cover and refrigerate at least 2 hours or up to 2 days. Pour mixture into the canister of an ice-cream freezer (I own this Cuisinart model and have been really happy with it!)

Freeze according to the manufacturer's instructions. Serve or store in freezer in an air-tight container.