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Sift the flour, baking powder, and salt together into a medium bowl.
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Beat the butter and shortening together in a large bowl with an electric mixer set to medium speed. When the mixture is very creamy, add add 3/4 cup sugar, the egg, the anise, and the vanilla, then beat to combine.
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Scrape down the sides and bottom of the bowl, then turn mixer to low and beat in the dry ingredients until just incorporated—don’t over-mix, the dough will be crumbly.
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With mixer still on low, beat in 1/4 cup of brandy. If necessary, continue adding brandy 1 tablespoon at a time to make a firm, cohesive dough.
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Divide the dough into 2 equal portions and wrap each with plastic wrap. Chill for 30 minutes (or up to 1 day).
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Preheat the oven 350 degrees. Line 2 large baking sheets with parchment paper or spray with cooking spray.
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Working with 1 portion of dough at at time, roll out to 1/4-inch thickness on a lightly floured work surface. Cut with a 2 1/2-inch cookie cutter.** Arrange cut-out cookies 2 inches apart on prepared baking sheets.
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Bake 10 to 16 minutes, until lightly browned. Cool on baking sheets for 1 minute, then transfer to a wire rack set on a large rimmed baking sheet (or any surface that’s easy to clean).
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While cookies are still hot, stir together the remaining 1/3 cup sugar and the cinnamon. Dust the hot cookies with the cinnamon sugar mixture, then let the cookies cool.
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Uneaten cookies can be stored in an airtight container at room temperature for up to 3 days or in the freezer for up to 3 months.