Beat the softened butter and granulated sugar together until light and fluffy, about a minute. Mix in the egg and molasses until fully combined and smooth.
Add the flour, ginger, cinnamon, baking soda, and salt. Mix until a thick dough forms. If the dough feels loose or sticky rather than thick, chill it for 5 to 10 minutes before scooping.
Scoop out 2 tablespoons of dough and roll it into a ball. Roll each ball in granulated sugar and place it on a parchment-lined baking sheet, spacing them a couple inches apart.
Bake at 350°F for 10 minutes, until the cookies are set at the edges but still look soft in the center.
Let the cookies cool and settle on the baking sheet for 15 to 20 minutes before moving them. They keep cooking from residual heat as they sit.
Whisk the powdered sugar, ginger, and water together until smooth. Dip the top of each cooled cookie into the glaze, then set it back on the tray or a wire rack until the glaze sets, about 15 minutes.