Using a food processor, combine the powdered flour, sugar, cocoa powder, and butter. Blend just until the mixture forms a ball.
Press the dough into the sides and bottom of a 9-10 inch tart pan. Pat to make the crust even.
Bake the crust until it is just cooked through, approximately 10 minutes. You can test it with a toothpick like a cake. Remove from oven and set aside to cool.
Make the filling by adding the cream cheese, sugar, vanilla, and peppermint extract in a small bowl. Beat until smooth.
Place all of the candy canes in a resealable bag or in a clean towel, covered. Hit the candy canes with a rolling pin or something heavy to crush.
Add half of the crushed candy canes and 1/4 cup white chocolate chips to the cream cheese mixture. Stir to combine. Adjust taste accordingly with more peppermint extract, candy canes, or vanilla.
When the crust is cooled down (slightly warm is okay, just not hot) add the cream cheese mixture and spread evenly throughout the tart.
Sprinkle the remaining crushed candy cane pieces, white chocolate chips, and sprinkles if desired across the tart.