Preheat oven to 325 degrees F. In a large saucepan combine sugar, coffee, unsweetened chocolate, and butter. Cook and stir over medium-low heat until chocolate and butter are melted; remove from heat and cool slightly.
Grease and flour a 10-inch fluted tube pan; set aside. In a small bowl combine flour, baking powder, baking soda, and salt; set aside.
Stir yogurt, eggs, and vinegar into the chocolate mixture in saucepan. Gradually add the flour mixture, stirring just until combined. Stir in 3/4 cup of the white baking pieces. Pour into the prepared pan.
Bake in the preheated oven for 50 to 60 minutes or until a toothpick inserted near the center comes out clean. Cool in pan on a wire rack for 15 minutes. Remove from pan. Cool completely on wire rack.
Place the remaining 1/2 cup white baking pieces in a small microwave-safe bowl. Microwave on 70 percent power (medium-high) for 1 minute; stir. Microwave on 70 percent power for 1-1/2 to 3 minutes more or until the chocolate is melted, stirring every 15 seconds. Repeat with milk chocolate or semisweet chocolate pieces, if using.
Place melted white mixture and, if using, melted chocolate into separate small plastic bags. Snip off a corner of each bag and drizzle each over the cake.