Preheat oven to 350 degrees F (175 degrees C). Line a muffin tin with greased liners.

Combine boiling water and cocoa powder in a small bowl; whisk until smooth. Allow to cool, about 10 minutes.

Sift flour, baking soda, baking powder, and salt together in a bowl.

Combine 1/2 cup butter and 1 cup plus 2 tablespoons white sugar in a large bowl; beat together with an electric mixer on medium speed until creamy. Beat in eggs one at a time until smooth. Mix in vanilla extract. Mix in the flour mixture in 3 additions, alternating with the cocoa mixture. Fold peppermint candies into the batter.

Scoop batter into the muffin tin.

Bake in the preheated oven until cupcakes are firm to the touch or a toothpick inserted into the center of a cupcake comes out clean, 15 to 18 minutes. Remove cupcakes from the pan and set on a rack to cool, about 30 minutes.

Combine shortening and 1/2 cup butter in a large bowl. Beat with an electric mixer on medium speed until smooth. Add peppermint extract. Add confectioners' sugar, 1/2 cup at a time, beating thoroughly after each addition. Beat in milk, a tablespoon at a time, until buttercream is desired consistency. Fold in food coloring a little at a time to reach desired shade of green.

Transfer buttercream to a piping bag fitted with a round or star tip. Frost each cupcake with buttercream.