1. In a bowl, with a mixer on medium speed, beat butter and brown sugar until blended. Beat in eggs. Add melted chocolate and vanilla; beat until combined.
2. In another bowl, stir together flour, cocoa, baking powder, baking soda, and salt. In a small bowl, mix milk and sour cream.
3. Add flour mixture and milk mixture alternately to butter mixture, about a third of each at a time, beating after each addition until incorporated (batter will be very thick). Scrape batter equally into two buttered and floured 9-inch round cake pans and spread level.
4. Bake in a 350° regular or convection oven until a wooden skewer inserted in the center comes out clean, 25 to 30 minutes. Cool on racks in pans for 10 minutes, then invert cakes onto racks and remove pans. Cool completely before frosting.
To melt the chocolate, break it into small pieces and place in a small microwave-safe bowl; cook in a microwave oven at half power (50%), stirring occasionally, until smooth, 2 to 3 minutes.