Place sliced bananas into a small baking dish. Top evenly with cubed butter and sprinkle with brown sugar. Bake for 20 to 25 minutes, until golden and bubbly. Remove and coarsely mash with a fork, still leaving chunks of bananas. Set aside.
Heat 3 inches of oil into a large Dutch oven or large deep-fryer over medium heat, to 350 degrees F.
In a large bowl sift the flour, cinnamon, baking powder, baking soda and salt; set aside. Gently whisk the heavy cream, sour cream and egg until just combined. Slowly stir banana mixture, cream mixture and cinnamon chips into dry ingredients until just combined.
In batches, using a medium cookie scoop (that holds approximately 1 1/2 tablespoons); scoop dough and carefully drop into hot oil, about 8 at a time. Fry for 3 to 4 minutes turning frequently, or until golden brown. Transfer doughnuts with a skimmer onto a paper towel lined plate to drain, until all have been fried.
In the top of a double boiler set over simmering water, mix the milk, butter, and vanilla until the butter is melted. Slowly add sugar, beating until glaze is smooth. Dip doughnuts into the glaze, turning to coat, and carefully transfer them to a large sheet of waxed paper to dry.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.