Sift flour, baking powder, and 3/4 teaspoon salt into a bowl. Cream butter and brown sugar with a mixer on medium speed until pale and fluffy, about 3 minutes. Add ginger, and beat for 2 minutes. Beat in egg and vanilla until well combined. Add flour mixture, and then beat on medium-low speed until dough just comes together.
Shape dough into two 1-inch-thick disks. Wrap tightly in plastic, and refrigerate for at least 1 hour (or up to 3 days).
Preheat oven to 350 degrees. Roll out 1 disk on a lightly floured surface to inch thickness, and cut out 2-inch circles. Place the rounds 1/2 inch apart on parchment-lined baking sheets. Roll out and cut scraps. Refrigerate rounds for 10 minutes.
Bake, rotating sheets halfway through, until cookies are set and edges are light gold, 12 to 14 minutes. Transfer parchment with cookies to wire racks. Let cool completely. Repeat with remaining dough. (Cookies can be stored for up to 3 days.)
Whisk confectioners' sugar, lemon zest and juice, honey, water, and a pinch of salt in a bowl until smooth. Drizzle glaze over the cooled cookies. Let stand until glaze is set, at least 20 minutes. (Glazed cookies can be stored overnight.)
Drizzling the rounds just before serving ensures a shiny glaze and crisp cookies. For softer cookies, let the glaze set overnight. (The glaze's shimmer will dull.)