Line two large baking sheets with parchment paper and set aside.

In a medium mixing bowl, combine the flour, baking powder, and salt and whisk to combine.

In a standing electric mixer fitted with the paddle attachment, cream the butter and sugar together until light and fluffy, scraping the sides of the bowl as necessary. Add the eggs one at a time, then add 1 ½ teaspoons of the anise extract, and mix until well-combined. With the mixer on low speed, add the flour in several increments, mixing only until just combined.

Using a small scoop or a spoon, portion dough into scant 2 tablespoon portions and, working with lightly floured hands, roll each scoop of dough into a smooth ball. Place onto the lined baking sheets, spacing them about 1 ½ inches apart, and refrigerate until well-chilled, at least 1 hour.

Preheat the oven to 350 degrees F.

Position oven racks in the upper and lower thirds of the oven. Bake the cookies, rotating sheets halfway through cooking, until very faintly colored on the bottom, about 18 to 20 minutes. Remove from the oven and allow to cool completely. Transfer cookies to wire racks.

In a small bowl, combine the powdered sugar with the remaining ½ teaspoon anise extract and the milk, adding enough milk to form a glaze slightly thicker than heavy cream.

One by one, dip the top of the cookies into the glaze, then return the cookies to the wire racks to dry. Sprinkle with the nonpareil sprinkles while glaze is still wet if desired. Cookies can be stored in airtight containers for up to several days once the icing has dried.

Makes about 2 ½ dozen cookies