Use an electric mixer to cream the butter and sugar on medium speed for 2 to 3 minutes, until smooth. Add the fresh ginger, powdered ginger, vanilla, flour, baking soda, and salt. Work in with a spatula to form a dough. If needed, work in 1 to 2 tablespoons of flour to bring the dough together.

On a lightly floured board, divide the dough into 4 portions. Roll each portion into a log about 1 inch (2.5 cm) thick. To make the dough easier to cut, cover and refrigerate until firm, about 20 minutes. Or, wrap each log and freeze for up to 3 months; thaw before using.

Line large baking sheets with parchment paper. Position a rack in the middle of the oven and preheat to 350 F (175 or 180 C).

Cut each log into scant ¼-inch (6 mm) coins; there should be 15 to 18 coins per log. Transfer to the prepared baking sheet, spacing them about 1 inch apart; as needed use your fingers to smooth out any cracks and maintain a round shape. Place and press a tiny piece of crystalized ginger in the center of each cookie.

Bake each sheet at a time for 10 to 12 minutes until the edges are lightly golden. Cool on the baking sheets set on wire racks. Stores very well in an airtight container.