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Beat together the butter, sugars, molasses, salt, and pumpkin pie spice until light and creamy.
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Beat in the egg, pumpkin purée, pumpkin flavor, and vanilla.
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Stir in the flour in three additions, mixing until the dough comes together.
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Divide the dough in half, shape both halves into disks, wrap in plastic, and refrigerate for at least 1 hour.
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Preheat the oven to 300°F. Grease two baking sheets, or line them with parchment.
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Working with one piece at a time, flour your work surface and roll the dough 1/4" thick.
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To shape cookies using a Springerle mold: Brush a very light coating of flour onto the dough and your Springerle mold. Press the mold firmly into the dough, then remove and cut around the design with a knife or a fluted pastry wheel.
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To shape cookies using a Springerle pin: Use a regular rolling pin to roll each disk of dough into a 1/4"-thick rectangle, roughly the same width as your Springerle pin. Use a pastry brush to brush a very light coating of flour onto the dough. Flour your Springerle pin, then give it a couple of sharp raps to knock off any excess. Slowly roll the Springerle pin over the dough, pressing down hard enough to leave a good impression. Cut the cookies apart on the lines, using a pastry wheel or knife.
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Transfer the cookies to the prepared baking sheets.
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Freeze the cookies for at least 30 minutes; this helps them retain their shape during baking.
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Bake the cookies for 30 minutes, until they're firm and dry looking; rotate the baking sheets halfway through. Remove the Springerle from the oven, and cool completely right on the pan.
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To make the glaze: Combine all the ingredients, stirring until smooth. Add enough milk to make a thin, pourable glaze.
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Dip the unstamped side of each cookie into the glaze; set the Springerle on a rack, and let the glaze set until firm.