Melt chocolate in saucepan on very low heat, stirring constantly; set aside.
Beat butter in a large bowl with electric mixer on medium speed until creamy. Gradually add about half of the powdered sugar, beating on low speed after each addition until well blended. Add chocolate and vanilla extract; mix well.
Add remaining sugar alternately with the milk, beating until well blended after each addition.
* For a variation* substitute 6 squares of Baker's semi-sweet baking chocolate for the 4 squares of Baker's unsweetened baking chocolate.