In a medium bowl, combine flour, baking soda and salt. Set aside.
In the bowl of a stand mixer, combine butter and sugar and beat until light and fluffy, about 3 minutes. Add the eggs, one at a time, beating well after each addition. Add vanilla. Then, add the flour mixture and beat until just combined.
Turn dough out onto a piece of plastic wrap and shape into a disc. Refrigerate the dough for at least an hour.
When ready to bake, preheat oven to 350 degrees.
Generously flour your work surface, remove dough from plastic wrap and place on flour. Flour rolling pin and roll dough to about 1/4-inch thick.* Cut to desired shape with cookie cutters and place them on a cookie sheet. Re-roll leftover scrapes and refrigerate any unused dough until ready to bake again.
Bake for 9-11 minutes until pale golden, let cool for 2 minutes, then transfer to a wire cooling rack (place a piece of foil under the rack for easy clean up later). Let cool completely before frosting. Repeat steps with the rest of the remaining dough.
To make the glaze, combine the confectioner's sugar and milk (start with 2 teaspoons). Add more milk or sugar to make the glaze as thick or thin as desired. Dip one side of the cooled cookies in the glaze and place back on rack (where the foil comes in handy!). Immediately top with sprinkles. Allow the glaze and sprinkles to set - the glaze will harden enough to stack the cookies when storing.
*Notes: the original recipe called for the dough to be rolled out between two sheets of wax paper. While this worked, after cutting out the shapes with the cookie cutters, the dough was sticking to the wax paper. I found that working on a floured surface was much easier.