For the cake, preheat oven to 350 degrees. Grease two 7-inch cake pans with vegan butter or cooking oil spray, and lightly dust with flour.
In a large mixing bowl, sift together flour, sugar, baking soda, and salt. In another mixing bowl, combine milk, oil, apple cider vinegar, vanilla and peppermint extracts. Add wet mixture to dry, and whisk until smooth. Divide cake batter among two cake pans, tapping against countertop to release bubbles.
Bake cakes for 28 to 30 minutes, or until a toothpick inserted into center comes out dry. Transfer to cooling rack and allow to reach room temperature. Once cooled, place in refrigerator until ready to assemble.
For the peppermint meringue buttercream, using a hand or stand mixer with whisk attachment at medium speed, add butter/shortening and whip until fluffy. Slowly add ½ cup sifted powdered sugar at a time to mixer while it runs, until all is incorporated. Add aquafaba and peppermint extract, whipping on medium-high until light and fluffy (about 5 minutes). Be careful not to get it too fluffy or it wont be firm enough for layering.
Onto a cake stand or large plate, place strips of parchment paper along edges of stand. Place 1 cake layer on stand, adjusting parchment paper along edges of cake (which will make frosting the cake easier and look cleaner). Pipe a ¾-inch-thick layer of white buttercream along top of first cake round, making sure there are no air pockets. Place next cake layer rounded side down on top of frosting. Using an offset spatula, spread a thin coat over any bare cake edges, carefully remove parchment paper, then refrigerate cake for 15 minutes.
Lightly dip a pastry scraper into warm water, then rotate cake while holding scraper straight with its edges, gently remove and even out excess frosting to give it a naked-cake look. Dip scraper back in water to clean as necessary. Once smooth, use offset spatula to place extra frosting on top of the cake, then decorate the outer edge with crushed and whole peppermint candies, as pictured. Carefully remove parchment and chill until ready to serve!