Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Pulse the hibiscus flowers 8 to 10 times in a spice grinder until coarsely ground but not powdery. Whisk 1 tablespoon of the ground hibiscus, the flour, ground ginger and 1/2 teaspoon salt in a medium bowl.
Grate the fresh ginger on the large holes of a box grater. Squeeze out the juice with your hands into a small bowl; discard the solids.
Beat the butter and 3/4 cup confectioners' sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 5 minutes. Beat in the vanilla, lime zest and 1 tablespoon of the ginger juice. Reduce the mixer speed to low; beat in the flour mixture in two batches until combined.
Roll tablespoonfuls of dough into balls. Arrange about 1 inch apart on 2 unlined baking sheets. Bake, switching the pans halfway through, until the cookies are light golden on the bottom, 15 to 18 minutes. Let cool 2 minutes on the pans, then transfer to racks to cool completely.
Meanwhile, make the glaze: Whisk the remaining 1 cup confectioners? sugar, 1 teaspoon ginger juice, the lime juice, water and 2 pinches of salt in a medium bowl. Dip the top of each cookie in the glaze to cover evenly; let any excess drip off. Sprinkle with the remaining ground hibiscus. Let set, about 5 minutes.