Preheat the oven to 350 degrees. Line a 9x9 metal baking pan with parchment paper; lightly spray with cooking spray.
In a large bowl, stir together the brownie mix, eggs, melted butter, and water. Pour about 2/3 of the brownie batter into the prepared pan, then sprinkle the chocolate chips over the top. Reserve the rest of the brownie batter to go over the cream cheese layer.
In another bowl, beat together the softened cream cheese and sugar until well combined and creamy. Add the flour and beat until combined. Add the egg and vanilla, beating agin until well combined, scraping down the sides of the bowl if needed.
Spread the cream cheese batter over the brownie batter. Drop the remaining brownie batter on top by spoonfuls. Use a toothpick to swirl the two batters together.
Bake at 350 degrees for approximately 35-38 minutes, until the edges are set and the top is just barely cracked, and a toothpick inserted near the center comes out mostly clean (a few crumbs are ok). If the toothpick comes out covered in chocolate that means you've hit a chocolate chip; test again somewhere else.
Allow the brownies to cool completely and then refrigerate 2 hours before serving. Serve chilled or at room temperature. Store leftovers in the fridge.