STIR flour with baking powder and salt in a medium bowl. Add cocoa powder and instant-coffee granules to flour mixture. Set aside.
BEAT butter with sugar in a large bowl, using an electric mixer on medium-high, until fluffy. Beat in egg and vanilla. Gradually beat in flour mixture, just until combined. Gather into a ball. Divide dough in half, then roll each portion into a 2-inch-wide log. Wrap logs in wax paper and twist ends.
CHILL until firm, 1 to 1-1/2 hours.
POSITION racks in top and bottom thirds of oven.
PREHEAT to 350F. Line 2 baking sheets with parchment.
SLICE cookies into 1/4-inch rounds and arrange 1 inch apart on prepared sheets. Bake in top and bottom thirds of oven, switching halfway through, until cookies are golden, 10 to 12 min. Let cookies cool on sheets for 1 min, then transfer to a wire rack to cool completely. Repeat with remaining dough. Cookies will keep well, stored in an airtight container at room temperature, up to 5 days.