Carefully pour into prepared pan. Bake until golden and knife inserted into center of batter comes out clean and cake springs back, about 50 to 60 minutes.
To make syrup: While cake is baking, combine 1/4 cup butter, water, mint leaves, sugar, and rum in a small saucepan. Heat to a simmer, then remove from heat and allow to cool. Once cool, strain mixture, pressing trapped liquid out of the mint. Discard mint leaves, set syrup aside.
To assemble: With the hot cake still in the pan, carefully level the top by cutting off the "dome" over the edge of the pan. Evenly pour the mint syrup over the entire cake. Allow cake to cool for about 20 minutes before turning cake out directly onto serving plate. This cake is delicate, and should not be moved to a different plate after being turned out!
To make icing: While cake is cooling on serving plate, heat remaining 1/4 cup amber rum. Add remaining mint leaves, and let steep for 10 minutes. Strain, once again pressing trapped liquid out of the mint. Discard mint leaves. Mix as much powdered sugar into the 1/4 cup rum as it takes to make a very thick paste. Melt it in the microwave for a few seconds, then drizzle over the cake. If desired, stir a tiny drop of green food coloring into the frosting before drizzling — pale mint green drizzle looks pretty on this cake!
Garnish with sprigs of fresh mint, if desired.
Nutrition information per serving: