Preheat oven to 350°F Line a 9x9x2-inch baking pan with foil, allowing edges to hang over sides of pan; set pan aside.
Beat butter and 1/4 cup shortening in a medium bowl with an electric mixer on medium speed for 30 seconds. Add brown sugar, granulated sugar, and baking soda. Beat until combined, scraping sides of bowl occasionally. Beat in egg and vanilla until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour. Stir in miniature semisweet chocolate pieces and, if desired, nuts.
Press dough evenly onto bottom of prepared baking pan. Bake in preheated oven for 25 to 30 minutes or until crust is evenly golden brown and center is set. Cool in pan on a wire rack for 1 hour. Reduce oven temperature to 325°F
Use foil to lift baked mixture out of pan. Transfer cookie to a cutting board; remove foil. Cut baked mixture into 9x1/2-inch slices using a serrated knife. Place slices, cut sides down, 1 inch apart on an ungreased cookie sheet. Bake in 325°F oven about 20 minutes or until crisp, turning carefully halfway through baking time. Cool completely on the cookie sheet on a wire rack. Trim ends, if desired.
Microwave chopped chocolate and 1 tablespoon shortening in a small microwave-safe bowl on 50 percent power (medium) for 2 to 3 minutes or until melted and smooth, stirring twice. Brush or spread one end of each cookie stick with melted chocolate mixture; let excess drip down sides of cookie. Place cookies on parchment paper or waxed paper; let stand about 1 hour or until set. Makes 18 cookies.
To store: Place cookies in layers separated by waxed paper in an airtight container; cover. Store at room temperature for up to 3 days or freeze for up to 3 months. Thaw cookies, if frozen, before serving.