Preheat the oven to 350°. Line 3 cookie sheets with parchment paper and set aside.
Whisk together the flour, baking powder, and salt in a medium bowl and put it to the side.
In a separate large bowl, add the lemon zest and sugar and mix them together with your fingers until fragrant. Add the butter, and use an electric hand mixer to cream the sugar and butter together until fluffy.
Mix in the eggs and vanilla extract.
Gradually add the dry ingredients and continue mixing until the dough is thick and sticky.
Use a 1 1/2 tablespoon cookie scoop to scoop out the cookie dough and roll it into smooth balls with your hands. Place the dough balls about 2 inches apart on the parchment-lined cookie sheets. (I put 12 cookies on each cookie sheet.)
Bake the cookies, 1 cookie sheet at a time, for 15 minutes or until the bottom edges of the cookies start to turn golden brown. Place them on a cooling rack to rest.
Add the powdered sugar, lemon juice, and milk to a small bowl and stir until it is smooth and free of lumps.
Once the cookies are cooled, dip the tops into the icing and place them on a piece of parchment paper or a cooling rack to let the icing run down the sides. Sprinkle with pastel-colored sprinkles and let the cookies rest for about 10 minutes so that the icing sets.
Notes
If your icing is too thin, add more powdered sugar. If the icing is too thick, add more milk. The icing should appear white, not clear.