For the cookies:
1 cup unsalted butter, softened
1/4 cup granulated sugar
1 teaspoon vanilla extract
2 cup all purpose flour
1/2 teaspoon salt
For the icing:
2 tablespoons milk
1 1/4 cup powdered sugar
1. Whisk flour and salt together in medium bowl; set aside.
2. In stand mixer fitted with paddle attachment, beat butter, sugar and vanilla at medium speed until light and fluffy, about 3 minutes, scraping down sides and bottom of bowl with rubber spatula. Reduce speed to low and add flour mixture, mixing until incorporated. Wrap tightly in the plastic wrap. Refrigerate for at least 1 hour.
3. Heat oven to 325 degrees. Line a baking sheet with parchment paper or a Silpat Baking Mat.
4. Roll a heaping tablespoon of dough into a 1½-inch ball, roll the ball in the sprinkles and place it on the prepared baking sheet. Repeat with the remaining dough, spacing the balls about 2 inches apart. Using your thumb, dowel or a melon baller, make an indentation in the center of each cookie. Bake for 10 minutes, or until slightly firm. Remove the cookie sheets from the oven. Using your thumb, dowel or a melon baller, press into the cookies again. Return the cookies to the oven and until golden brown, 8 to 13 minutes. Remove cookies to a wire rack and allow to cool completely before icing.
5. To make the icing, whisk the milk and powdered sugar together in a small bowl and add some food coloring. Stir to combine and add more milk (up to 2 teaspoons more) to get a smooth and spoonable, but not runny, consistency. Fill centers with icing. Let set overnight. Cookies can be stored in airtight containers up to 1 week.