In a large bowl, whisk together milk, oil, granulated sugar, eggs, and anise extract until well-combined. Add flour, baking powder, and salt, and mix with a wooden spoon until smooth. Set aside and let batter rest for 10-15 minutes.
Preheat oven to 350 degrees F with rack in the upper-middle position. Line three baking sheets with parchment paper. Drop cookie batter by rounded teaspoonfuls onto the baking sheets, about 1-1/2 inches apart. (I use a 1-1/2 teaspoon cookie scoop.)
Bake, one sheet at a time, for 10 minutes, until centers of cookies bounce back when gently pressed. The cookies should not brown. Remove from oven and let rest 2-3 minutes.
While cookies are still warm, remove from baking sheets by gently lifting each cookie while peeling back the parchment paper (a thin crumb impression of the cookies will remain on the parchment; this is normal.) Transfer cookies to wire racks to cool before glazing.**
Make the glaze
In a medium bowl, whisk together confectioner's sugar, milk, salt, and anise extract or liqueur until smooth (I use liqueur). When cookies have cooled to room temperature, dip tops into glaze and sprinkle immediately with nonpareils. Let stand at room temperature until glaze is dry.
*My grandmother's cookies, made with 3 teaspoons (1 tablespoon) of extract always had a lovely, prominent anise flavor. I love them this way. If you prefer a more subtle flavor, or if you're using an artisanal brand of extract that's particularly potent, decrease the quantity in the batter to 1-1/2 to 2 teaspoons.**I place a layer of wax paper under my cooling racks to make cleanup after glazing a breeze. When finished, just roll up the paper, with all of the dripped glaze and sprinkles, and discard.The longer the cookies sit, the color of the nonpareils might "bleed" a little into the glaze. This is normal and does not affect flavor. These cookies should not be frozen and are best enjoyed within a day or two of baking.