Preheat the oven to 400 degrees. Line a baking sheet with parchment paper.
Whisk flour, sugar, ginger, cardamom, baking powder, baking soda, and salt together until thoroughly combined. Grate in frozen butter using a box grater. Toss until mixture resembles coarse crumbs. Mix in chopped ginger.
Whisk buttermilk, egg, and vanilla extract together in a small bowl or measuring cup until thoroughly combined. Add to dry ingredients and mix just until no dry clumps of flour remain; dough will be somewhat sticky.
Turn dough out onto the prepared baking sheet. Gently press dough together into an even 7-inch circle. Lightly flour your hands if dough is too sticky to shape. Cut dough into 8 triangles using a bench scraper. Space scones 1 inch apart on the baking sheet.
Whisk egg and water until combined. Gently brush the top of each scone with the egg wash. Sprinkle with coarse sugar.
Bake in the preheated oven until scones are lightly browned, 20 to 25 minutes. Cool on the baking sheet for 5 minutes before transferring scones to a wire rack to cool completely.
Yields 8