In a large bowl, beat the butter and sugar with an electric hand mixer until smooth and creamy. Add the vanilla and beat until well mixed.
Stir in the flour and salt until well mixed. Form the dough into a square, wrap in plastic wrap, and chill 20-30 minutes.
Once chilled, roll the shortbread between two layers of parchment paper (one on the bottom, one on top of the shortbread dough) until about 1/2 inch thick. You should have roughly a 10x8 inch square.
Remove the top layer of parchment and cut the dough into 2x1.5 inch squares (you'll have 24). Gently transfer the parchment paper with the dough onto a cookie sheet.
Whisk the egg white and water until well mixed and brush on top of each cookie. Sprinkle with granulated cane sugar.
Bake until the edges are just golden brown, about 12-14 minutes (they set up a lot!). Cool completely on the pan.