Preparation

Cookies

1. In a stand mixer or a medium bowl with an electric beater, beat butter and sugar for 5 minutes, until very fluffy. Add eggs one at a time and beat on medium until smooth; mix in salt and vanilla. Reduce the speed of the mixer and incorporate the flour.
2. Divide the dough in four pieces. Put a sheet of parchment on the counter, then form each portion of dough into a 1/2-inch thick rectangle; place on parchment. Cover with another sheet of parchment and roll out to 1/8- or 1/4-inch thickness. Stack the rolled-out dough on a sheet pan, with each layer separated by parchment paper, and wrap the whole thing tightly. Chill for at least 2 hours and up to 3 days.
3. Set 2 oven racks in the upper and lower thirds of the oven. Preheat to 350⁰F.
4. Take out a sheet of dough and lay the dough on a cutting board or kitchen counter. Remove top parchment, cut with 3-inch cutters, and place cut cookies on parchment-lined pans with 2 inches between. Press the remaining scraps of dough together and roll out again; chill as you bake the previous batch.
5. Bake for 5 minutes, switching the position of the pans between upper and lower oven racks and bake for 5-8 minutes, until golden around the edges. Cool on racks on parchment.
6. Repeat the rolling, chilling, cutting, baking and cooling process until all the dough is used up.

Frosting

1. Put the butter in a stand mixer or a large bowl with an electric mixer, and beat to soften. Add the powdered sugar and stir on low until it is almost mixed. Add the vanilla with the machine running, and drizzle in the milk gradually while mixing. When the mixture is smooth, whip for a few seconds to add fluffiness. If it's too thick to pipe, gradually drizzle in milk, a teaspoon at a time.
2. Transfer the frosting to a piping bag, or zip-top bag with a corner cut off, and decorate the cooled cookies. Let the frosted cookies dry on racks for 10 minutes before transferring to platters or storage containers (for up to 3 days).