Portion the dry ingredients (flour, baking powder, salt) in a bowl, set aside.
Portion the wet ingredients (egg, vanilla and milk) in another bowl, set aside.
Cream the butter and sugar in your mixer on a medium speed. Add the wet ingredients to the butter mixture in three stages. Mix until fully incorporated for each stage. When the last of the wet is incorporated, turn the mixer off.
Add the dry ingredients all at once. Mix on speed 1 just until the dough starts coming together. Give it a 5 second flash on a medium-high speed, to insure the mixture is homogeneous. Wrap the dough in plastic wrap in a puck shape. Chill for 1-2 hours.
Preheat oven to 350 degrees F. Prepare a baking sheet with a Silpat or parchment paper.
On a lightly floured surface, roll out the dough to 1/4 inch thickness. Cut into desired shapes with cookie cutters. Place cookies 1 inch apart on the prepared cookie sheets. Bake for 8-12 minutes in the preheated oven, until bottoms and edges of cookies are light brown. Remove from baking sheet and cool on wire racks.
Melt the butter and water together over a double boiler. When the butter is melted, remove the double boiler from the heat, but keep the double boiler in tact. Add the powdered sugar all at once. Stir with a whisk, pressing the whisk against the sides of the bowl to press out the lumps. When all your lumps are worked out, add more water, 1 teaspoon at a time (if needed) until the glaze reaches your desired thickness. Add food coloring as desired.
In a mixer, whip the egg white in a clean bowl until it starts to get foamy with soft peaks. Add the powdered sugar all at once. Whip on a slow speed until fully combined. Use immediately or cover with plastic wrap directly on the surface of the icing.
To decorate the cookies:
Dip the tops of your cookies in the warm glaze and allow them to fully dry before applying royal icing. If adding sprinkles or sugar, be sure to sprinkle on while glaze is still wet. Likewise, you can pipe a border of royal icing onto the cookie, allow to fully dry, and then flood the center with warm glaze.
Serving size based on one cookie. Estimated total quantity of recipe is 4 dozen cookies, however that can vary depending on thickness of dough and size of cut outs.