Preheat the oven to 350° F, with racks in the middle and lower third of the oven. Have two nonstick cookie sheets available.

Place the flour, salt, and sugar in a food processor bowl. Process for a few seconds to mix the dry ingredients.

Cut the butter into 1/2-inch chunks and scatter on top of the flour. Pulse-process until the butter is cut into the flour so that the mixture looks like coarse meal.

In a small bowl, combine the vanilla extract, water, and egg. Whisk lightly to blend. With the processor running, add the egg mixture through the feed tube. Continue to process until the dough just starts to clump together. If the dough is too dry to come together, add more water, 1 teaspoon at a time. The dough won't ball up in the bowl, but it should start to clump, and should come into a ball with the heat of your hands as you push it together. Wrap the dough in plastic wrap and let it rest for 15 minutes.

Roll the dough (I like to roll in a floured jumbo zip top bag) to a scant 1/8 inch thick. Cut with a dreidel-shaped cookie cutter, and place on the cookie sheets. Bake for 5 minutes, switch the cookie sheets top to bottom, and bake another 4 to 7 minutes, until the edges are just starting to brown. Remove the cookies from the pan, and cool them on a wire rack.

For the icing, sift the powdered sugar Into a medium bowl. Whisk in the milk and vanilla until smooth. The icing should be very thick but thin enough to spread. If it is not thin enough, add more milk by the 1/2 teaspoonful until the icing is the right consistency. Spread the cookies with icing (a small decorating spatula works best). If you are going to use sprinkles, sprinkle them on while the icing is still wet so they will stick well. If not using sprinkles, let the icing dry. Add color to the remaining icing (I like to use a rich blue) and paint on the dreidel letters using a fine artist's brush. For a great-looking accent, sprinkle small, fine sprinkles, such as multicolored tiny balls, on the wet lettering (larger sprinkles, such as chocolate sticks, will obscure the letters). If you prefer, the dreidel letters can be painted on un-iced cookies. In all cases, let the decorations dry before storing the cookies.

Cookies can be made 1 day ahead, or frozen for up to 3 months.