In a medium bowl, whisk together flour, baking powder, salt, cinnamon, nutmeg, and ginger. Set aside.
In a large bowl or stand mixer, cream together 2/3 cup butter*. Add ¼ cup packed brown sugar and ¼ cup white sugar. Beat for about 2 minutes, scraping the sides well.
Add egg and molasses. Beat on high until it's not lumpy anymore.
Use a food processor or a rolling pin to crush 7 candy canes. Add crushed candy and dry ingredients all at once and mix until just combined, scraping sides.
Line a baking sheet with parchment paper or a silpat. Use a 2-inch cookie scoop to drop them on the sheet about 2 inches apart. Bake at 350 degrees for 11-14 minutes. Take them out when they are no longer shiny on top.
Remove to a wire rack after a couple minutes and let cool completely.
Meanwhile, cream the butter in a large bowl. Add the powdered sugar, vanilla, and milk and combine well, scraping the sides of the bowl. Add more milk if it is too thick, powdered sugar if it is too thin.
Crush 2-4 candy canes in a food processor or with a rolling pin.
When the cookies have cooled, frost the cookies and immediately sprinkle with crushed candy canes. Or if you want you can stir that candy cane into the frosting, then spread it on. These will keep on the counter in a sealed container for 2 or 3 days.
Notes
*I know 2/3 cup butter is kind of an awkward measurement. It's just one stick of butter, plus 2 and 2/3 tablespoons. Use the measurements on the wrapper to figure this out, it's not too bad.You don't need to refrigerate this dough, but if you wait a long time in between batches I would stick it in the fridge just so they don't get melty on you when you bake them.Source: adapted from my best friend's grandma, Dorothy Millikin