STIR flour with cocoa, baking powder and salt in a medium bowl.
BEAT 1 cup butter with granulated sugar in a large bowl using an electric mixer until light and fluffy, 3 min. Beat in vanilla and egg. Gradually beat in flour mixture just until combined. Divide dough in half. Shape each half into a flat disc and wrap separately with plastic wrap. Refrigerate until firm, at least 2 hours, or over-wrap with heavy foil and freeze up to 1 month.
POSITION oven racks in top and bottom thirds of oven. Preheat oven to 375F. Spray 2 baking sheets with oil. Dust a rolling pin with flour. Roll a disc of dough no thicker than 1/8 in. Cut out shapes with a 2-in. round, fluted cookie cutter. Transfer to prepared baking sheets and place about 2 in. apart. Bake in top and bottom thirds of oven, switching and rotating sheets halfway through, until cookies are slightly puffed, about 8 min. Repeat with remaining dough. Gather dough scraps and form into a disc. Chill until firm enough to re-roll, about 10 min.
BEAT butter in a large bowl using an electric mixer until light and fluffy, 2 min. Alternately beat in sugar and cream, starting and ending with sugar. Stir in peppermint candies. Fill a piping bag fitted with a large tip or a small zip-top plastic bag (snip off a small corner). Pipe 1 tbsp of peppermint filling onto half of cookies and top with remaining cookies. Best served day of baking. (Shopping Tip: Little round peppermint candies work just as well as candy canes. To crush them, place in a bag and smash with a rolling pin.)