Step 1

Heat the oven to 350 degrees F.

Step 2

To make the cake: Cut the banana peels into 1-inch pieces and place them in a small saucepan with 1 cup of water. Bring to a boil over medium-high heat, then reduce the heat and simmer for 10 minutes. Remove the pan from the heat and allow the mixture to cool slightly, then drain the banana peels, reserving 1/4 cup of the cooking water.

Step 3

Meanwhile, butter and flour the sides of two 8-inch round cake pans and line the bottoms with parchment paper. Butter and flour the pans again to coat the paper.

Step 4

Transfer the peels and the 1/4 cup of cooking water to a tall, narrow container and puree with an immersion blender until completely smooth (a mini food processor would do the trick, too!).

Step 5

Cream together the butter and sugar in a large bowl using an electric mixer (or a wooden spoon for an arm workout) until pale and fluffy, 3 to 5 minutes. Add the egg yolks one at a time, mixing until incorporated, and scraping down the sides of the bowl after each addition. Mix in the banana peel mixture, then stir in the buttermilk until well-combined.

Step 6

In a medium-size bowl, whisk together the flour, baking soda, baking powder, and salt. Add the dry ingredients to the bowl with the butter mixture and stir gently, just until combined.

Step 7

Put the egg whites in another bowl (make sure it's clean and dry!) and whisk until soft peaks form -- either by hand or with the whisk attachment on an electric mixer. If using an electric mixer, start slowly and gradually increase speed to medium-high. You're done when you pull out the whisk or beater and a soft peak is formed, but immediately collapses. Gently fold the egg whites into the batter and divide the batter evenly between the two prepared pans.

Step 8

Bake, rotating the pans halfway through, until the tops are golden and a toothpick inserted into the center of each cake pulls out with dry crumbs rather than wet batter, about 25 minutes. Let the cakes cool completely in the pans.

Step 9

When the cakes are completely cool and you're ready to assemble, make the frosting: Melt the butter in a medium saucepan over low heat. Stir in the brown sugar and cook, stirring constantly, for 2 minutes. Stir in the milk, raise the heat to medium-high and cook, stirring constantly until the mixture boils. Remove from the heat and let cool to lukewarm. Gradually whisk in 1 3/4 cups powdered sugar, beating until smooth. Add the additional 1/4 cup powdered sugar if the frosting is too loose. Use the frosting immediately, as it will begin to thicken and stiffen as it sits.

Step 10

To remove the cake from the pans, invert one cake pan on a serving plate, lift pan, and peel the parchment. Repeat for the second cake pan. Put one layer of the cake on a serving platter and spread about one-third of the frosting evenly over the top. Set the other layer on top, and spread the remaining frosting over the top and sides of the cake.