1. In a food processor or bowl, whirl or stir flour, sugar, and salt to blend. Add butter; whirl or cut in with a pastry blender until mixture resembles coarse meal.
2. In another bowl, whisk the egg, vanilla, lemon peel, and lemon juice to blend. Add to flour mixture in food processor or bowl; whirl or stir until the dough forms a ball.
3. Divide dough into thirds. Roll each portion, between sheets of waxed paper or cooking parchment, into a round 1/8 inch thick. Stack and freeze rounds until firm, about 30 minutes (or chill for about 2 hours).
4. Cut out shapes with a floured, 2-inch, 5-point star cutter. Space cookies 1/2 inch apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets.
5. Bake cookies in a 325° oven until pale golden at tips, 8 to 10 minutes. Transfer to racks to cool. Reduce oven temperature to 150°.
6. In a 1-quart pan over medium heat, stir jam until melted. Push through a fine strainer, discard residue, and return liquid to pan. Bring to a simmer and cook for 2 minutes to thicken, stirring occasionally. Let cool just until warm to touch, 7 to 8 minutes. Meanwhile, in a bowl, stir 1/2 cup plus 2 teaspoons powdered sugar and 4 teaspoons water until smooth.
7. Brush cookies with jam, then place slightly apart on 12- by 15-inch baking sheets. With a clean pastry brush, paint a thin film of powdered sugar glaze over each cookie to completely cover jam (beginning with the cookies coated earliest with jam).
8. Bake cookies until glaze is set and dry, 15 to 18 minutes. Transfer to racks to cool.
9. In a small bowl, stir remaining 1/2 cup powdered sugar and the egg white until smooth. To assemble cookies, spoon a dot (1/8 teaspoon) of powdered sugar mixture in center of one cookie, then gently press another cookie on top, with star points offset.
10. Let cookies stand until icing is set, about 1 hour. Serve, store airtight up to 2 days, or freeze.
Nutritional analysis is per double cookie.