Preheat the oven to 350 degrees F. Butter and lightly flour a 2-quart ramekin or souffle dish and tap out the excess flour. Put the ramekin on a baking sheet.
Combine the flour, baking powder, baking soda, salt, cinnamon, allspice and ginger in a large bowl. Whisk the butter, granulated sugar, brown sugar, banana, sour cream, eggs and vanilla in a separate large bowl until smooth. Whisk the flour mixture into the banana mixture until just combined.
Pour 2 1/2 cups of the batter into the ramekin and stack the ice cream bars on top of each other in the middle. Pour the remaining batter around the sides and over the top of the ice cream bars.
Bake until the cake is set and the center is still jiggly, about 45 minutes. Let the cake cool in the ramekin for 15 minutes. Dust with confectioners' sugar and scoop out portions with a spoon to serve.
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