Cream butter and sugar, vanilla, and egg.
Gradually beat in flour, baking powder, and salt until blended.
Press dough together with hands. Roll out to 1/8- to 1/4-inch thickness, adding flour as needed to avoid sticking.
Cut into desired shapes.
Bake at 375 on ungreased cookie sheets for around 7 minutes. Don't overcook them (they'll be crunchy if overcooked, and perfectly soft if you nail the timing), and take them out of the oven when they look slightly undercooked, and haven't paled on the top yet. They'll be slightly golden on the bottom when they're done. Cooking times will vary greatly depending on how thick you cut your cookies.
For the icing, beat powdered sugar into the egg white until a nice spreadable consistency is reached. Add food dye if desired* and ice the cookies only when they're cool.