Sift together flour, sugar and salt. Cut butter into small pieces and drop them into the flour mixture. Using a pastry knife or your own cold hands, work butter into flour until the mixture resembles coarse oatmeal and no butter pieces are larger than pea-sized. Add cold milk and vanilla extract and gently knead it until the dough forms a ball. Flatten the dough into a disk, wrap in plastic wrap, and chill 1 hour.
Preheat oven to 350 degrees. On a lightly floured board, roll dough out to a 12 inch round. Transfer to tart pan and press into the bottom and up the sides to just higher than the edge of the pan. Pierce the bottom with a fork to prevent bubbles. Freeze for 20 minutes to re-harden butter. Bake empty crust until lightly browned, about 30 minutes. Flatten any bubbles and transfer to rack to cool.
Bring cream and corn syrup to simmer in medium saucepan then remove from heat. Add chocolate and whisk until smooth. Cool 30 minutes. Pour filling into crust. Sprinkle with 1/2 cup candy. Chill until set, about 3 hours. Crust and filling can be prepared a day ahead and kept refrigerated.
Combine cream, powdered sugar, and peppermint extract in a large bowl or the bowl of a stand mixer. Beat until whipped cream forms stiff peaks. Spoon over chilled tart, mounding slightly in the center. For a more decorative tart, use a pastry bag or large Ziploc bag with the corner snipped off to pipe whipped cream in patterns. Sprinkle remaining tablespoon of crushed candy canes over the whipped cream. Chill at least 30 minutes; serve chilled.
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