Place about 2 inches apart on baking sheets. Bake in a 300-degree oven until cookies no longer feel soft when gently touched but are not browned, about 18 minutes. Cool on pans about 5 minutes. Place remaining powdered sugar in cake pan. Gently roll hot cookies, a few at a time, in sugar to coat well.
Return cookies to paper in a single layer.
Shake cocoa through fine strainer over cookies. Let cookies cool. Serve, or store airtight at once for up to 3 days. Freeze to store longer.