Peppermint Meringue Cake with Chocolate Buttercream

Position rack in center of oven and preheat to 175 °F. Using pencil, trace two 12 x 4-inch rectangles on sheet of parchment paper; turn parchment over (rectangles will show through) and place on large rimmed baking sheet. Trace one 12 x 4-inch rectangle on second sheet of parchment paper; turn parchment over and place on second rimmed baking sheet. Sift powdered sugar, superfine sugar, and salt into medium bowl. Combine egg whites and cream of tartar in large bowl of heavy-duty stand mixer fitted with wire whisk; beat on medium-low speed until frothy. Increase speed to medium-high and beat until soft peaks form. Increase speed to high; gradually add sugar mixture and beat until stiff and glossy peaks form, about 3 minutes. Beat in vanilla and peppermint extracts. Divide meringue equally among prepared rectangles on parchment (about 1 1/3 cups for each); spread meringue evenly within traced lines. Bake meringues until dry and crisp but not browned, about 3 hours. Turn off oven and let meringues cool completely in oven. Carefully remove meringue rectangles from parchment. DO AHEAD: Meringues can be made 1 week ahead. Store in single layer in airtight container at room temperature.