In a large bowl using an electric hand mixer or stand mixer, cream together the butter and sugar on low speed until fluffy and light in color.
Stop the mixer and scrape down the sides of the bowl with a rubber spatula. Add 1 egg and the vanilla extract, and beat until combined.
With the mixer on low speed, add the flour and salt and mix until just combined. Scrape down the sides of the bowl again to ensure the dough is evenly mixed.
Remove the dough from the bowl and shape it into a smooth rectangle. Place between 2 sheets of parchment paper and roll into a rectangle about 9x12 and ¼ inch thick. Place the dough, still in the parchment paper, on a baking sheet. Freeze until firm for about 20 minutes, or refrigerate for about 2 hours.
Meantime, preheat the oven to 350F and prepare the egg wash. In a small bowl, combine the egg and 1 teaspoon water.
Remove the dough from the freezer/fridge and remove the top parchment paper. Using a cookie cutter or cookie stamp of your choice, cut out your cookies, placing each one evenly on a parchment-lined baking sheet. Brush with egg wash and sprinkle with sugar.
Bake for 12-14 minutes until firm and lightly golden. Let the cookies cool for about 5 minutes on the baking sheet before serving.