Make cake: Preheat oven to 350 degree F and arrange racks in upper and lower thirds; butter 3 round 9-by-2-inch cake pans. Line bottoms with rounds of parchment or waxed paper and dust pans with flour, tapping out excess.

Whisk boiling water into cocoa in a heatproof bowl until smooth. Add chocolate and let stand 1 minute. Whisk until smooth; let cool 5 minutes. Whisk in buttermilk and extract.

Whisk together flour, baking soda and salt in another bowl. Beat butter and sugars together in a large bowl on medium speed until light and fluffy, about 4 minutes. Add eggs, one at a time, beating well and scraping down sides of bowl with a spatula after each addition. Reduce speed to low; alternately add flour and chocolate mixtures, beginning and ending with flour mixture, beating until just combined. Divide batter evenly between prepared pans (about 4 cups each) and smooth tops.

Bake until a toothpick inserted in center of each cake comes out with a few crumbs clinging to it and cakes begin to pull away from sides of pan, 40 to 45 minutes. Transfer cakes in pans to a rack and let cool 5 minutes.

Run a thin knife around edges of pans and place racks over pans; invert cakes onto racks. Peel off paper and invert cakes again so they are right side up on racks; cool completely.

Make glaze: Meanwhile, bring cream to a simmer in a small saucepan over moderate heat. Remove from heat; add chocolate and butter. Stir until smooth, about 5 minutes, and transfer to a bowl. Let cool 15 minutes.

Make filling: Meanwhile, place whole peppermint candies and confectioners' sugar in a food processor; pulse until finely ground. Transfer to a bowl and stir in cream cheese, butter, and extract.

Assemble cake: Place 1 cake layer on an 8-inch cardboard round set on an 8-inch round wire rack. Spread with half of filling and cover with second cake layer, right side up. Spread with remaining half of filling and top with remaining cake layer, right side up. Place cake on rack in a baking pan and pour some glaze over top, smoothing onto sides with a small metal spatula. Continue to pour and spread glaze, using drippings in pan if necessary, until cake is evenly coated. Chill cake on rack until glaze is just set, about 15 minutes. Garnish with chopped candies around top edge of cake, if desired.

Transfer cake on cardboard round to a cake stand or plate. Bring to room temperature before servings. Makes 12 servings.