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To make the cookies: In a medium-sized bowl, beat together the sugar, butter, and salt.
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Beat in the egg, water, and vanilla, then the flour and cocoa.
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Divide the dough in half and pat each half into a rectangle. Wrap well and refrigerate for 1 hour (or longer).
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Preheat the oven to 325°F. Lightly grease (or line with parchment) one, two, or three baking sheets (however many you have).
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Once the dough has chilled, take one piece out of the refrigerator. Flour a clean work surface.
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Roll the dough 1/8" thick. If it starts to stick, add additional flour to both the top and bottom of the dough. Alternatively, place the dough on parchment and put a sheet of plastic wrap over it as you roll, pulling the plastic to eliminate wrinkles as necessary when rolling; this will keep the dough from sticking without the need for additional flour.
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Cut out 2" circles from the dough and transfer the cookies to the prepared pans, spacing them 1" apart.
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Gather the scrap dough, roll, and repeat. If at any time during this process the dough becomes sticky and hard to work with, simply refrigerate it for about 20 minutes, until firm.
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To shape linzer-style cookies: Once you've transferred the cookies to a baking sheet, use your smallest seasonal cookie cutter or the end of a round piping tip to make a cutout in the center of half of them.
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Bake the cookies until you can smell chocolate and the tops and edges are set, 18 to 20 minutes.
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Remove the cookies from the oven and allow them to cool completely, on a rack or on the pan.
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To make the filling: Beat together the shortening, sugar, vanilla, and peppermint until clumpy.
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Add the water, beating until smooth and stiff; the filling should be spreadable. If it feels too stiff, add more water 1/4 teaspoon at a time.
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To assemble traditional cookies: Place a level tablespoon of filling onto the bottom of one cookie — a teaspoon cookie scoop, slightly heaped, works well here. Spread the filling out with your fingers or a small spatula, then place another cookie atop the filling, bottom side down. Press the cookies together gently. Repeat with the remaining cookies.
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To assemble linzer cookies: Place a level tablespoon of filling onto the bottom side of the solid (uncut) cookies. Spread the filling to the edges, then top with the cookies with cutout centers, bottom side down.
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To garnish with crushed peppermint candies, fold the candies into the filling before using. Or roll the outside edge of the finished sandwiches in the candy so it sticks to the filling.
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Storage information: Store cookies in an airtight container for several days.