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1
In a small saucepan combine the apricots, 1/4 cup of the granulated sugar, and 2/3 cup of water, simmer the mixture for 15 to 18 minutes, or until the mixture is reduced by half, and add the rum.
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2
Let the mixture cool slightly and in a blender puree it.
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3
In a bowl whisk together the flour, the baking powder, and a pinch of salt.
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4
In a large bowl with an electric mixer beat together the butter, the remaining 1/2 cup granulated sugar, and the brown sugar until the mixture is light and fluffy, beat in the vanilla and the egg, and beat the mixture until it is combined well.
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5
Add the flour mixture and beat the dough until it is just combined.
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6
Form the dough into a log, chill it for 1 hour, and divide it into 8 pieces.
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7
Working with 1 piece of dough at a time, on a sheet of plastic wrap form the pieces into 8-inch ropes, wrapping the plastic wrap around the ropes to keep the dough from sticking, and on 2 baking sheets pat the rope into a 8- by 1 1/2-inch rectangles.
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8
Make a canal down the center of each rectangle with your finger and spread the apricot puree in the canals.
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9
Bake the rectangles in 2 batches in the middle of a preheated 350F.
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10
oven for 18 to 20 minutes, or until the edges are golden, transfer them to racks, and let them cool.
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11
In a small bowl whisk together the confectioners' sugar, the lemon juice, and enough water to make a thick but pourable icing, drizzle the icing over the rectangles diagonally into 1-inch strips.
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12
The cookies keep in an airtight container for 1 week.
