Plantain Cake with Avocado Buttercream

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2 cups superfine sugar

2 3/4 cups cake flour

1/2 teaspoon ground cinnamon

1/2 teaspoon salt

1 tablespoon baking powder

1 cup warm milk

1 very ripe plantain (mostly black)

1/2 cup unsalted butter

1/4 cup shortening

4 large eggs

1 1/2 teaspoons vanilla extract

Avocado Buttercream, recipe follows

Plantain Chips, recipe follows

Avocado Buttercream:

1 cup unsalted butter

7 cups powdered sugar, divided

2 ripe avocados

1 tablespoon vanilla extract

1/4 teaspoon salt

Plantain Chips:

1 plantain

Sugar, as needed