Use an electric hand mixer on medium speed (or a stand mixer outfitted with the paddle attachment) to mix the butter for about 10 seconds. Pause to add the sugar. Beat for 3 to 4 minutes until pale colored and aerated. Remove the beaters (or paddle).
Add the fresh ginger, ground ginger, vanilla, baking soda, salt, and flour. Use a spatula to combine. If it's dry, crumbly, and doesn't come together, add 1 teaspoon water. Transfer onto a lightly floured work surface, then gather and gently knead dough into a ball.
Divide into 4 portions, rolling each into a generous 1-inch-thick log. Wrap each in parchment paper then plastic wrap. Label with masking tape, and freeze for up to 3 months. If the cookies are to be baked immediately, cover then chill the dough for 20 minutes, which will make it easy to slice.
To bake, heat the oven to 350°F with a rack in the middle position. Line a baking sheet with parchment paper.
Partially or fully thaw the dough log before using. Cut it into 1/4-inch-thick slices (about 16 total). Place the flat rounds 1 1/2 inches apart on the prepared baking sheet. Press a crystallized ginger cube in the center of each. Bake at 350°F for 10 to 12 minutes, until spread out, lightly puffed up, and lightly golden at the edges. Cool cookies on the baking sheet on a wire rack for 8 to 10 minutes before eating.