Cream together the butter and sugar in a stand mixer/food processor or with a hand mixer until smooth and slightly glossy. Add the egg and vanilla and blend in.
Add the ginger, baking powder, flour and cornstarch and mix in at a low speed to save the flour going everywhere. Bring the mixture together as a ball and chill for around 30 minutes.
Preheat the oven to 375F/190C.
Roll out the chilled dough fairly thin on a floured surface with a floured rolling pin. Use cutters (shapes, circles, as you prefer) to cut out cookies and transfer to a lined baking sheet/tray/cookie sheet. Repeat with all the dough.
Bake for approx 10 minutes until just starting to brown. Transfer to a cooling rack to cool completely.
Sift the confectioners sugar into a small bowl and mix with the cocoa powder and water. Blend until smooth, then either spread on the cookies with a flat knife for a simple glaze, or transfer to a small piping back (or small freezer bag then snip off a corner) and decorate as you choose. Note if you glaze more fully, you may need a little more glaze.