In a large mixing bowl, cream together the butter and sugar. Then stir in the eggs and vanilla until just combined.
In another mixing bowl, whisk together the flour, salt, and baking powder. Then stir the dry ingredients in with the wet until a dough forms.
On a floured surface, roll the dough out so it's 1/4-inch thick or a little less. Cut out shapes with cookie cutters. Place on a baking sheet lined with parchment paper or a baking mat.
Bake at 350°F for 6-8 minutes. The centers may look a little undercooked, but the edges should just be beginning to brown. Remove to a cooling rack and they will firm up.
To make the royal icing, simply whisk together the meringue powder, powdered sugar, and water. Start with five tablespoons and add an additional one (or two) as needed. The mixture should flow in thick ribbons, but not be super runny. Use a piping bag or ziplock to pipe the frosting onto the cooled cookies.
The cookies are best if left to set overnight, but you can also consume them immediately if you like. The frosting will get harder as it sets, so this is best if you plan to transport the cookies anywhere.