Add potato and mix gently with a fork. Cut in butter until pieces are the size of small peas.
Sprinkle extract over mixture, a few drops at a time, mixing lightly with fork (dough should be crumbly). Shape dough into a ball; chill thoroughly.
Roll dough, a half at a time, 1/4 inch thick on a floured surface; cut with 1 3/4 inch round cutter and cut centers from HALF the rounds with a 3/4 inch round cutter. Place rounds, rings, and centers on separate ungreased cookie sheets.
Bake at 400°F 10 to 15 minutes.
When cool, spread rounds with currant jelly and top with rings; fill centers of rings with a small amount of jelly; top with centers. About 2 dozen cookies.