In a large mixing bowl, beat butter with an electric mixer on medium to high speed for 30 seconds. Beat in granulated sugar and vanilla. Add flour; beat until combined. Divide dough in half. Cover and chill for 2 to 24 hours.
On a lightly floured surface, roll each half of dough to 1/4-inch thickness. Cut with a 2-1/2- to 3-inch diamond-shaped cutter, or cut with a knife or pastry wheel into diamonds. Place cookies on an ungreased cookie sheet. In a small bowl, mix egg yolk and milk. Brush onto tops of cookies; sprinkle lightly with bread crumbs.
Bake in a 375 degree F oven for about 10 minutes or until edges are lightly browned. Remove from oven; cool on wire racks.
Spread jam on half of the cookies. Top with remaining cookies. Sift powdered sugar over cookie sandwiches. Makes about 24 tea cookies.