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Combine butter and powdered sugar in a large bowl. Beat at medium speed until creamy.
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Stir in vanilla.
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Reduce speed to low; add flour and salt.
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Beat until well mixed.
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Divide the dough in half and flatten each half into a disk shape.
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Wrap the dough in plastic wrap.
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Refrigerate until firm (1 hour).
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Heat oven to 300°F.
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Roll out the dough on a lightly floured surface, one-half at a time (keeping remaining dough refrigerated), to 1/8-inch thickness.
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Cut dough with 1 1/2-inch round cookie cutter.
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Place them 1 inch apart onto ungreased cookie sheets.
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Whisk the egg white with water in small bowl and brush over surface of cookies. Sprinkle with sugar.
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Bake for 10 to 12 minutes or until set; cool completely.
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Meanwhile, heat apricot preserves in a small saucepan over medium-low heat until mixture just begins to bubble (1 to 2 minutes).
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Continue cooking 1 minute.
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Strain to remove large pieces, if necessary.
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Cool completely.
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Spread a thin layer of preserves onto unsugared-side of cookie; top with second cookie, sugared-side down.
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Repeat with remaining cookies.