Heat oven to 325°. Beat butter and sugar with a hand mixer until fluffy. Add vanilla and egg; beat. Add flour and salt; mix. Transfer dough to a floured surface, form into a disk, roll until 1⁄4" thick. Using a 2" fluted round cutter, cut out cookies and transfer to parchment paper-lined baking sheets, spaced 2" apart. Chill for 30 minutes. Using a 1 1⁄4" plain round cutter, cut out centers of half the cookies; discard centers. Bake cookies until lightly browned, 10-12 minutes; cool.
Dust cookie rings with confectioners' sugar. Place 1 tsp. jam in center of each whole cookie; spread to within 1⁄3″ of edge. Top each with a cookie ring.