In the bowl of a stand mixer, cream together the butter, sugar, and lemon zest until light and fluffy. Add the egg and vanilla; beat to combine.
In a separate bowl, combine the flour, baking powder, and salt. Add to the butter mixture and combine. Continue to beat until the dough comes together, pulling away from the sides of the bowl.
Divide the dough in half and roll out on a floured surface to about 1/4-inch thick. Cut into shapes and transfer to a parchment-lined baking sheet. Refrigerate cut-out cookies on the baking sheet for 15 to 20 minutes, or until cold.
Meanwhile, preheat the oven to 350° F.
Bake cookies for 9 to 10 minutes, or until set and light brown around the edges. Let sit on the baking sheet for a couple of minutes before transferring to a wire cooling rack.
Lemon Royal Icing
Combine the ingredients in a bowl and stir until smooth.
Add water as necessary, about 1/2 teaspoon at a time, until the icing is the right consistency. I like to run a spatula through the icing. For me, I like the consistency when it takes about 10 seconds for the ripples to disappear and the icing is flattened.
Dip each baked cookie, one at a time, in the frosting by holding onto the edges and dipping in just the surface. Let the excess icing drip back into the bowl and then quickly flip the cookie right side up and lay it on a flat surface.
Using a toothpick, you can guide and spread frosting into any gaps. Let the icing dry before storing. Depending on how thick the frosting is, this could take several hours.
Notes
Icebox Variation: This dough works great as an icebox cookie too. Instead of forming the dough into a disk, form it into a log, wrap it in plastic, and refrigerate until ready to bake. Then, simply slice and bake.