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Preheat oven to 350 degrees F (175 degrees C). Line a muffin tin with paper liners. Mix chestnut flour, 1/2 cup plus 2 tablespoons white sugar, vanilla sugar, and baking powder together in a large bowl. Whisk 10 tablespoons melted butter and eggs together in a small bowl. Pour butter mixture into flour mixture; gently fold together until batter is just moistened. Scoop batter into muffin tin, filling each liner 2/3 of the way full. Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 20 minutes. Remove from oven; transfer muffins to a wire rack to cool. Heat a large skillet over medium heat. Add banana slices in a single layer. Sprinkle 1 tablespoon white sugar on top and flip the slices. Cook until golden brown, 3 to 5 minutes. Flip again and sprinkle remaining 1 tablespoon white sugar on top. Continue cooking until second side is golden brown, 3 to 5 minutes more. Mix cream cheese and 2 1/2 tablespoons softened butter together in a bowl. Add 1/2 cup plus 2 tablespoons confectioners' sugar and vanilla extract. Whip vigorously until icing is smooth and reaches spreading consistency, 2 to 3 minutes. Slice muffin tops off with a paring knife. Place a slice of caramelized banana on each muffin and cover with tops. Frost muffins with cream cheese icing. Top each one with a slice of caramelized banana.