In mixing bowl with electric hand-held mixer, cream butter and sugar until light and smooth. Beat in vanilla and melted chocolate. Stir in flour, mixing gently until thoroughly blended. Spread batter into a greased 9x13x2-inch baking pan.
Bake at 300° for 40 to 45 minutes, until center is set. Cool in pan and cut into 1 1/2- x 2-inch bars. Carefully remove chocolate shortbread bars with a metal spatula.
Makes about 36 chocolate shortbread cookies.