Preheat and prep: Preheat oven to 350°F. Line two baking sheets with parchment paper. In a medium bowl, whisk together flour and cocoa powder. Reserve a handful of chopped chocolate, then stir the rest into the flour mixture.
Cream butter and sugars: In stand mixer with paddle attachment, cream butter, brown sugar, granulated sugar, salt, baking soda, baking powder, and vanilla. Beat on medium speed for 5 minutes until light and fluffy. Scrape down sides of bowl as needed.
Add egg: Add egg to butter mixture and beat until completely smooth and incorporated.
Combine ingredients: Pour in dry ingredients all at once and mix on low speed just until dough comes together. Do not overmix.
Shape and bake: Using a 2-tablespoon cookie scoop, portion dough and roll into balls. Place on prepared baking sheets 2 inches apart. Flatten each ball into a disc about 1/2 inch thick. Press reserved chocolate pieces onto top of each disc. Bake for 13-15 minutes until edges are set but centers look slightly underdone.
Cool: Let cookies cool on baking sheet for 5 minutes, then transfer to a wire rack to cool completely before dipping.
Melt coating: Place white chocolate and coconut oil in microwave-safe bowl. Microwave in 30-second intervals, stirring between each, until completely smooth.
Dip and decorate: Dip each cooled cookie halfway into melted white chocolate, letting excess drip back into bowl. Place on parchment-lined baking sheet. Immediately sprinkle crushed candy canes over wet white chocolate.
Set: Let cookies sit at room temperature for 30 minutes until white chocolate coating firms up completely. Refrigerate for 10 minutes if needed to speed setting.