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