1. Preheat the oven to 375 degrees F. Line two baking sheets with parchment paper or a silicone baking mat.
2. In a medium bowl, whisk together the flour, baking powder and salt; set aside.
3. Cream the butter, shortening and granulated sugar on medium speed until light and fluffy, scraping the bowl as needed, about 3 minutes. Add the eggs one at a time, beating well after each addition. Mix in the anise extract.
4. On low speed or with a wooden spoon, mix in the flour and half-and-half alternately, beginning and ending with the flour.
5. Using 1-2 tablespoons of dough, roll into balls (or use a medium-sized cookie scoop) and place on the prepared baking sheets about 2 inches apart. Flatten slightly with your fingers. Bake for about 10 minutes, or until just lightly browned. Remove to cooling racks to cool completely before glazing.
6. For the glaze, whisk together the melted butter, powdered sugar, milk and anise extract until smooth. Dip the top of each cookie into the glaze, letting the excess drip off, and set right-side-up on the cooling rack. The glaze will harden once it dries. Store in an air-tight container at room temperature.
* Note 1: The amount of baking powder is not an error - this is what creates the cake-like texture of the cookies.
* Note 2: For an even more potent anise flavor, substitute 1 teaspoon of anise oil for the 1 tablespoon of anise extract in the cookies.