Line a 9×13-inch baking pan with a parchment sling and lightly coat the sides with nonstick spray. Preheat the oven to 350°F.
Whisk the pudding mix with the cold milk and 1/4 teaspoon peppermint extract for about 2 minutes until smooth. Refrigerate until slightly thickened but still pourable, 10–15 minutes.
Prepare the cake batter according to the box directions, then mix in 1/2 teaspoon peppermint extract. Spread the batter into the prepared pan, smoothing the top.
Bake 24–28 minutes, or until a toothpick inserted near the center comes out with a few moist crumbs. Cool on a rack for 5 minutes (cake should still be warm).
With the handle of a wooden spoon or a thick chopstick, poke deep holes all over the cake about 1 inch apart, reaching the bottom of the pan. Gently twist the tool to widen the holes slightly for better absorption.
Give the chilled pudding a quick whisk. Pour half over the warm cake, guiding it into the holes with an offset spatula. Firmly tap the pan on the counter to help it settle, then pour on the remaining pudding and tap again.
Refrigerate the cake uncovered for 30–40 minutes, until the pudding is set and the cake is fully cooled to the touch.
For the topping, chill a mixing bowl and whisk or beaters for 5 minutes. Beat the heavy cream to soft peaks, then sift in the hot chocolate mix and add the vanilla and a pinch of salt. Continue beating to medium-stiff peaks (creamy and sliceable).
Spread the hot-chocolate whipped cream over the cold cake. Chill 15 minutes for cleaner slices.
Warm the hot fudge until pourable and drizzle over the cake. Finish with crushed peppermint candies just before serving, then slice and enjoy.