Beat butter and brown sugar at medium speed with an electric mixer until blended; add eggs, beating well. Stir in melted chocolate, 1 teaspoon vanilla, and next 3 ingredients. Pour into tart shell.
Bake at 375° for 20 minutes. Cool on a wire rack 30 minutes.
Beat whipping cream at high speed until foamy; gradually add sugar, beating until soft peaks form. Stir in remaining 1 teaspoon vanilla. Serve with warm tart.