Step 1

Whisk flour, baking powder, and salt in a medium bowl. Using an electric mixer on medium speed, beat eggs, granulated sugar, butter, oil, and 1½ tsp. anise extract in a large bowl until smooth, light, and foamy, about 2 minutes. Reduce mixer speed to low and add dry ingredients; beat just to combine.

Step 2

Roll tablespoonfuls of dough into balls with your hands. Space at least 1" apart on 2 large parchment-lined baking sheets. Chill until very firm, about 1 hour.

Step 3

Arrange racks in upper and lower thirds of oven; preheat to 350°. Bake cookies, rotating sheets halfway through if coloring unevenly, until golden brown around the edges and tops are firm, 10–12 minutes. Let cool.

Step 4

When cookies are cool, whisk powdered sugar, remaining 1 tsp. anise extract, and 3 Tbsp. water in a large bowl until smooth (glaze should be the consistency of heavy cream; add water by the teaspoonful to loosen if needed). Gently toss cookies in glaze until lightly coated, letting excess drip back into bowl. Transfer to clean parchment-lined baking sheets (you can use a small offset spatula for this, but the glaze is pretty forgiving of any finger marks) and let sit until glaze hardens, about 2 hours.

Step 5

Do Ahead: Cookies, without glaze, can be made 2 months ahead; store airtight and freeze.