- Stir flour with baking powder and salt in a medium bowl. Set aside. Add 3 tbsp each finely chopped red and green glacé cherries and candied orange and lemon peel and 1 tsp pumpkin-pie spice to flour mixture.
- Beat butter with sugar in a large bowl, using an electric mixer on medium-high, until fluffy. Beat in egg. Add 1 tsp rum extract with egg. Gradually beat in flour mixture, just until combined. Gather into a ball. Divide dough in half, then roll each portion into a 2-in.-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.