In a bowl, with an electric mixer on medium speed, beat butter, granulated sugar, and vanilla until smooth. Stir or beat in flour and salt until dough is no longer crumbly and comes together in a ball. Transfer to a lightly floured surface.
Divide dough in half. With a lightly floured rolling pin, roll out each portion to about 1/8 inch thick. With a floured 2- to 3-inch round or star-shaped cutter, cut out cookies. If desired, cut a 1/2-inch circle out of the center of half the cookies to make a window for the jam filling (see notes). Place cookies 1 inch apart on buttered or cooking parchment- lined 12- by 15-inch baking sheets. Gather excess dough into a ball, reroll, and cut out remaining cookies.
Bake cookies in a 325° oven until edges are just beginning to brown, 10 to 12 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. With a wide spatula, transfer cookies to racks to cool completely.
Spread 1/2 tablespoon jam on the flat side of each of half the cookies (those without holes). Top each with a remaining cookie (with a hole), flat side toward jam. Sprinkle tops of cookies evenly with powdered sugar.
Note: Nutritional analysis is per sandwich cookie.