Preheat oven to 350°F and butter a 9x13" metal baking pan.
In a medium bowl, stir together the flour, sugar, baking powder, ginger and salt and blend in the butter, working it in with your fingertips until the mixture resembles coarse meal. Press evenly into the bottom of the pan (it will be thin) and bake until golden and crisp, 20 to 25 minutes.
Just before the shortbread is done, melt the butter in a small saucepan and whisk in the golden syrup, icing sugar, ginger and vanilla. Bring to a boil and simmer for 30 seconds.
Remove shortbread from oven and pour topping over, tilting pan and using a spatula to cover it evenly. Cool in the pan, then cut into small rectangles or break into shards.