Preparation
With a long serrated knife, cut the cake crosswise into 2 layers. Put the layers, cut side up, on a small baking sheet. With a grapefruit knife, carve out enough cake to make a 1" deep by 1 1/4" wide cavity around the center of each layer, leaving a scant 1/2" shell.
To make the filling, mix the frozen yogurt, crushed candy canes, and food coloring in a large bowl with a rubber spatula until well blended. Spread evenly in each cake shell and freeze until firm, about 1 hour.
To assemble the cake, top the wider layer with the remaining layer, cut side down. Spread the top, center hole, and sides with the whipped topping. Freeze until the topping is firm, about 3 hours. Cover the cake with plastic wrap; then wrap in heavy-duty foil. Freeze at least 8 hours or 1 week.
To make the glaze, whisk the brown sugar and cocoa in a small saucepan until blended. Gradually whisk in the evaporated milk until smooth. Cook, whisking constatly, over medium-low heat until the mixture comes to a simmer, about 4 minutes. Cook, whisking constantly, until the mixture is slightly thickened and shiny, 1-2 minutes more. Remove the saucepan from the heat;stir in the vanilla and the peppermint eaxtract (if using). If making ahead, let the glaze cool and transfer to an airtight container. Cover and refrigerate up to 1 day. Let return to room temperature before using.
To serve, unwrap the cake and transfer to a serving plate. Drizzle the glaze evenly over the top of the cake and sprinkle with chopped candy canes (if using). Let stand at room temperature until slightly softened before slicing, about 15 minutes.