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Make the dough: Into a bowl sift together the flour and the sugar, add the butter, and blend the mixture until it resembles meal. Add the egg yolk, the rum, and 2 tablespoons ice water, toss the mixture until it is combined, and form the dough into a ball. Knead the dough lightly with the heel of the hand against a smooth surface for a few seconds to distribute the fat evenly and re-form it into a ball. Dust the dough with flour and chill it, wrapped in wax paper, for 30 minutes.
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Halve the dough, roll out half of it 1/8 inch thick on a floured surface, and cut it into rounds with a 1 1/4-inch fluted cutter. (The scraps may be chilled, re-rolled and cut into more rounds.) Bake the rounds on a baking sheet lined with parchment paper in a preheated 325° F. oven for 12 minutes, or until the edges are faintly golden. Transfer the cookies to a rack and let them cool. Sandwich the cookies together with the apricot jam. Makes about 35 cookies.
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Roll out the remaining dough 1/8 inch thick on a floured surface and cut it into rounds with a 1 1/4-inch round cutter. Cut out the centers from half the rounds with a 1/2-inch round cutter. (The scraps may be chilled, re-rolled, and cut into more rounds.) Bake the rounds and the rings in a preheated 325° F. oven for 12 minutes, or until the edges are faintly golden. Transfer the cookies to a rack and let them cool.
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Spread the rounds lightly with some of the ganache, cover each one with a ring, and fill the centers with a rosette of ganache piped through a pastry bag fitted with a 3/8-inch fluted rosette tip. Makes about 45 cookies.
