Bring first 3 ingredients to a boil in a small saucepan over medium heat. Cover, reduce heat, and simmer 10 minutes. Remove from heat; let stand 10 minutes.
Process apricot mixture, 1 cup sugar, butter, and vanilla extract in a food processor 1 to 2 minutes or until smooth, stopping to scrape down sides. Cool.
Roll 1 piecrust to press out fold lines. Cut 6 (4-inch) circles from pastry, rerolling as necessary. Brush edges of circles with water. Spoon about 2 tablespoons apricot mixture in center of each circle. Fold circles in half, pressing edges to seal. Place on a lightly greased baking sheet. Repeat procedure with remaining piecrust and filling. Stir together egg and water; brush over pies. Sprinkle evenly with 3 tablespoons sugar.
Bake at 425° for 20 minutes or until golden.
*1/4 cup water and 1/4 teaspoon almond extract may be substituted for almond liqueur.