Preheat oven to 325ºF (160º C). Grease an 8×8 inch baking pan, or line with parchment paper.
In the large bowl of a standing mixer, beat melted butter with sugar, cocoa and salt until well combined. Add eggs, one at a time, beating after each addition. Beat in vanilla extract. Add flour and beat to combine. Immediately transfer to prepared baking pan, reserving ½ cup of batter.
For cream cheese swirl: In the bowl of a standing mixer, beat together cream cheese, sugar, egg and flour. Drop by spoonfuls, alternating with spoonfuls of the reserved brownie batter, into pan. Use a butter knife to gently swirl chocolate and cream cheese batters together.
Bake brownies in centre of oven for about 30 minutes, until cream cheese batter is set and toothpick inserted in centre comes out almost clean. Allow brownies to cool in pan.