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1
Simmer apricots, water, and 1/4 cup granulated sugar in a small saucepan, stirring until sugar is dissolved and liquid is reduced by about half, 12 to 15 minutes.
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2
Remove from heat and add rum, then cool slightly.
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3
Puree in a food processor or blender until smooth (use caution when blending hot liquids).
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4
Whisk together flour, baking powder, and a pinch of salt in a small bowl.
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5
Beat together butter, brown sugar, and remaining 1/2 cup granulated sugar in a large bowl with an electric mixer at moderate speed until pale and fluffy, about 5 minutes.
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6
Beat in vanilla and egg.
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7
Add flour mixture and mix at low speed until just combined.
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8
Form dough into a log and chill, wrapped in plastic wrap, until firm, about 1 hour.
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9
Preheat oven to 350F.
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10
Divide dough into 8 pieces.
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11
Put 1 piece of dough on a sheet of plastic wrap, then fold plastic wrap over dough.
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12
Form piece into an 8-inch rope (dough will be very soft).
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13
Transfer in plastic wrap to 1 of 2 ungreased baking sheets, then remove plastic wrap.
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14
Pat rope into an 8- by 1 1/2-inch rectangle.
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15
Make more ropes and pat into rectangles in same manner, arranging them about 2 inches apart.
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16
Make a shallow trough lengthwise down center of each rectangle with your finger and spread apricot puree in trough.
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17
Bake in batches in middle of oven until edges are golden, 18 to 20 minutes, then transfer to a rack to cool completely.
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18
Whisk together confectioners sugar, lemon juice, and enough water (about 2 1/2 teaspoons) to make a thick but pourable icing.
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19
Drizzle icing over rectangles and cut diagonally into 1-inch-thick strips.
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20
Let stand until icing is set, about 15 minutes.
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21
Cookies keep, layered between sheets of wax or parchment paper, in an airtight container at room temperature 1 week.
