Preheat oven to 325°. Put flour, butter, granulated sugar, and salt in the bowl of a stand mixer. Mix on low speed to blend, then increase to medium and mix until dough is no longer crumbly and just comes together.
Form dough into a disk and chill 30 minutes. On a lightly floured work surface, roll out dough until 1/8 in. thick. Use a selection of 1 1/2-in. decorative cutters to cut as many shapes as you can, making sure you have an equal number of each shape to form a top and bottom and rerolling scraps as needed. Arrange cookies 1 in. apart on baking sheets. Use a variety of smaller cutters to remove center from half of cookies (the tops). Chill on sheets 15 minutes.
Bake until light golden brown, 12 to 15 minutes.
When cool, spread each whole cookie with about 1/2 tsp. jam. Sprinkle half of cut-out tops evenly with 1/2 cup powdered sugar. Mix remaining 1 cup powdered sugar with milk to thin, then dip tops of remaining cut-out tops with glaze. If you like, dip some glazed tops into sparkle sugar. Set all tops on jam-topped bottoms.