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Make cookies:
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Whisk together flour, baking powder, salt, cardamom, cinnamon, and allspice in a bowl.
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Beat together butter and sugar in a large bowl with an electric mixer at medium-high speed until pale and fluffy, about 2 minutes in a stand mixer (preferably fitted with paddle attachment) or 4 minutes with a handheld. Beat in eggs and vanilla. Reduce speed to low, then mix in flour mixture until just combined.
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Form dough into 2 (12-inch) logs (1 1/2 inches in diameter), each on its own sheet of plastic wrap. Use plastic wrap and your hands to roll, press, and square off sides of logs. Chill logs on a baking sheet until slightly firm, about 1 hour, then smooth logs with plastic wrap and flat side of a ruler to achieve straight sides. Chill logs on baking sheet until firm, about 1 hour.
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Put oven racks in upper and lower thirds of oven and preheat oven to 350u00b0F.
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Cut enough scant 1/4-inch-thick slices from a log with a knife to fill 2 large ungreased baking sheets, arranging slices about 1 inch apart (chill remaining dough, wrapped in plastic wrap).
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Bake cookies, switching position of sheets halfway through baking, until edges are golden, 10 to 12 minutes total. Cool on sheets 3 minutes, then transfer to racks to cool completely. Make more cookies with remaining dough on cooled baking sheets.
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To make Icing for cookies:
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Whisk together espresso powder, vanilla, and 1 1/2 tablespoons milk until espresso powder is dissolved, then add confectioners sugar and enough additional milk to make a thick but pourable icing. Spoon into a sealable bag and snip 1/8 inch off a bottom corner.
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Spoon melted chocolate into another sealable bag and snip 1/8 inch off a bottom corner.
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Pipe some espresso icing and chocolate over each cookie and let cookies stand on racks until icing sets, about 2 hours.
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Cooks' notes:
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Dough logs can be chilled 5 days or frozen, wrapped in a double layer of plastic wrap, 1 month (thaw in refrigerator just until they can be sliced).
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Cookies (with or without icing) keep, layered between sheets of wax paper or parchment, in an airtight container at room temperature 1 week.
