the bowl of an electric mixer fitted with the paddle attachment.
Add butter and granulated sugar; mix on medium speed until fluffy. Mix in vanilla and egg. Reduce speed to low. Add flour mixture; mix until combined. Halve dough; shape into disks. Wrap in plastic; refrigerate until firm, at least 2 hours.
Preheat oven to 375 degrees. Working with 1 disk at a time, roll out dough on a lightly floured surface to 1/8 inch thick. Refrigerate 20 minutes. Cut out squares with a 2-inch fluted cutter. Cut out centers of half the squares with a 1/2-inch heart cutter; re-roll scraps. Space 2 inches apart on parchment-lined baking sheets. Bake squares and hearts until pale golden, 8 to 10 minutes. Transfer to racks to cool.
Meanwhile, heat jam in a small saucepan over medium heat until reduced and thickened, about 7 minutes; let cool.
Sprinkle cutout cookies with confectioners’ sugar. Spread jam onto uncut squares; top with cutout ones. Store in an airtight container up to 2 days.