Whisk together flour, baking powder and salt until blended; set aside. In another large bowl, use electric mixer to beat together cocoa powder, oil and ground coffee until combined. Beat in both sugars. Add first egg; beat in completely. Add second egg and vanilla; beat in completely. Stir in dry ingredients just until blended. Cover dough; chill 4 hr. (or overnight).
Preheat oven to 350°F (180°C). Line two baking sheets with parchment paper. Using a tablespoon measure, scoop dough into heaped spoonfuls (the dough will be sticky). Roll into balls then roll in icing sugar to coat; shaking off excess. Place balls on prepared baking sheets about 2 in. (5 cm) apart.
Place sheets on top and bottom-third oven racks, switching racks and rotating sheets halfway through baking. Or use the middle rack and bake cookies one sheet at a time, rotating sheet halfway through baking. Bake cookies 6 to 8 min. (total bake time) until surfaces begin to crack. The interior of the cookies will look molten. Transfer cookies to wire rack to cool completely. Store in airtight container up to 3 days or freeze up to 1 month.
For cocoa crinkle cookies, roll in cocoa powder instead of icing sugar.
Sandwich coffee ice cream between 2 cooled cookies. Freeze until ice cream is firm before serving.