In a large mixing bowl, beat butter, powdered sugar, egg, and vanilla on low speed until combined. Increase speed to medium and beat until creamy, about 1 minute.
Add cocoa powder, all-purpose flour, baking powder, and salt to the wet ingredients. Mix on low until just combined.
Wrap dough in plastic wrap and chill until firm. 20-30 minutes in the freezer should do it. Roll dough into an 8-inch log and chill in the fridge until firm, about 1 hour.
Preheat oven to 375 degrees F and line two baking sheets with parchment paper or silicone baking mats.
Make sure cookie dough log is round by rolling it gently before slicing. Slice dough into 1/4-inch thick rounds and place on lined baking sheets. Space them about 1-inch apart, they will spread.
Bake 8-9 minutes, rotating sheets halfway through baking. Remove from oven and cool completely on wire racks
When cookies are cooled, melt white chocolate chips and vegetable oil in a wide-mouthed microwave-safe bowl. Heat chocolate for 30 seconds at 50% power, and then stir. If needed, repeat heating for 15 seconds at a time and then stir until smooth. (see note!)
Place crushed peppermints in a shallow, wide-mouthed bowl. Dip a cookie into the white chocolate to coat 1/2 the cookie and then immediately dip the chocolate side in crushed peppermints. (You can also sprinkle the crushed candy onto the white chocolate.)
Continue working with one cookie at a time until all cookies are coated with white chocolate and candy. Set dipped cookies back on the wire rack to dry and let the chocolate harden. Store in an airtight container.
Notes
Be sure to always microwave on 50% power. White chocolate can be fussy. If it gets too hot it will seize. Also, be sure you are using fresh white chocolate chips. Chips that aren't fresh won't melt properly.