Preheat oven to 350 degrees. Grease a foil-lined 9-inch by 13-inch baking dish. Prepare the brownie mix according to the package directions and pour into the prepared pan. Bake until top of brownie is shiny and center is almost set, about 15 minutes.
Meanwhile, in a standing mixer fitted with the paddle attachment, beat the cream cheese, sugar and vanilla on medium speed until well mixed. Add sour cream and mix thoroughly. Add the eggs, one at a time, mixing on low speed until blended. Pour the mixture over the partially baked brownie, filling the mixture to almost the top of the pan.
Bake until center is almost set, about 40 minutes. Remove from oven and run a knife or metal spatula around the rim of the baking dish to loosen cake. Cool and then refrigerate 4 hours or overnight. Allow to stand 30 minutes at room temperature prior to serving. Remove from the pan using the foil as a handle.
Drizzle with chocolate and let stand until chocolate is firm. Cut into 16 pieces and serve.