Preheat your oven to 375°F and line two baking sheets with parchment paper. In a medium bowl, whisk together the all-purpose flour, cocoa powder, baking powder, and salt until well combined.
In a large mixing bowl, beat the softened butter and granulated sugar with an electric mixer on medium speed for 2-3 minutes, until the mixture is light and fluffy. Add the egg and vanilla extract, and beat just until fully incorporated.
Gradually add the dry flour mixture to the wet ingredients, mixing on low speed just until a soft, uniform dough forms with no visible streaks of flour. The dough will be slightly sticky.
Place the powdered sugar in a shallow bowl. Scoop out 1 tablespoon of dough, roll it between your palms into a 1-inch ball, then roll it generously in the powdered sugar until completely coated.
Arrange the coated dough balls 2 inches apart on the prepared baking sheets. Bake for 8-10 minutes, until the cookies are set around the edges but still soft in the center. They will develop characteristic crinkled tops.
Let the cookies cool on the baking sheet for 5 minutes to firm up, then transfer them to a wire rack to cool completely.