Slice bananas into ½ inch pieces and toss them with brown sugar and butter in a baking dish. Bake at 400F for 35-40 minutes, stirring once, until bananas are browned.
Scrape bananas and their syrup into a blender. Add remaining ingredients and puree until smooth. Chill.
Stir the very thick mixture to loosen it, then churn in your ice cream maker.