Preheat oven to 375°. In a microwave, melt ⅔ cup chocolate chunks; stir until smooth. Cool slightly. In a large bowl, cream butter and confectioners' sugar until light and fluffy, 3-4 minutes. Beat in sweet potatoes, cooled melted chocolate, ginger and vanilla. In another bowl, whisk flour, cornstarch, baking cocoa, baking powder, baking soda and salt; gradually beat into creamed mixture.
Shape dough into ¾-in. balls; place 2-½ in. apart on parchment-lined baking sheets. Flatten slightly with bottom of a glass dipped in confectioners' sugar. Bake until edges are firm, 8-10 minutes. Remove from pans to wire racks to cool completely.
Meanwhile, mix filling ingredients until smooth. Spread filling on bottoms of half the cookies; cover with remaining cookies.
For chocolate coating, microwave remaining chocolate chunks; stir until smooth. Dip cookies halfway into chocolate or drizzle chocolate over tops of cookies; let stand until set. Store between pieces of waxed paper in an airtight container in the refrigerator.
