The delicate blance of sweet, bitter and citrus that makes the cocktail so refreshing also makes for an unusual citrus cake.
Prepare the Cake: Preheat oven to 325°F. Grease three (8-inch) round cake pans with baking spray. Process sugar and orange peel in a food processor until peel is finely chopped and resembles the texture of wet sand.
Beat together butter and orange sugar in the bowl of a stand mixer fitted with paddle attachment until light and fluffy. Add eggs, one at a time, beating after each addition.
Whisk together flour, baking soda, and salt in a separate bowl. Stir together orange juice and milk in a 2-cup measuring cup. Add flour mixture to butter mixture alternately with milk mixture, beginning and ending with flour mixture. Beat on low speed just until combined, scraping down sides as needed.
Pour batter evenly among prepared cake pans, and bake in preheated oven until a wooden pick inserted into cake comes out clean, 35 to 40 minutes. Cool cake in pans on a wire rack at least 20 minutes before inverting them to wire rack to cool completely. Place one cake layer on a cake stand, and brush with Aperol.
Prepare the Frosting: Beat together powdered sugar and butter in the bowl of a stand mixer, until light and fluffy. Add orange juice, Aperol, and a drop of each food coloring, and beat until smooth and desired color.
Spread one cup of frosting on top of cake layer, and spread to the edges of the cake. Top with second cake layer, and repeat process. Top with remaining cake layer, and spread remaining frosting on top and sides of cake until smooth and cake is fully frosted. Garnish with orange slices and mint.