Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Place one-third of the white chocolate in a small microwave-safe bowl; microwave in 30-second intervals, stirring, until melted. Set aside to cool slightly. Whisk the flour, baking powder and salt in a medium bowl. Beat the butter and sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 2 minutes. Beat in the melted white chocolate until smooth, then beat in the egg and vanilla. Reduce the mixer speed to low; add the flour mixture in 2 batches and beat until combined. Beat in half of the remaining chopped white chocolate. Roll heaping tablespoonfuls of dough into balls and arrange about 2 inches apart on the prepared baking sheets. Bake the cookies, switching the position of the pans halfway through, until the edges are slightly golden, about 15 minutes. Let cool 3 minutes on the baking sheets, then transfer to a rack to cool completely. Make the glaze: Microwave the remaining chopped white chocolate (you can use the same bowl as before) in 30-second intervals, stirring, until melted; stir in the heavy cream, and peppermint extract to taste. Dip the tops of the cookies in the glaze and return to the rack. Sprinkle with the crushed candy cane and let set, about 1 hour.