Make brownie batter: Put oven rack in middle position and preheat oven to 350°F Butter an 8-inch square baking pan. Heat butter and chocolate in a 3-quart heavy saucepan — though I did mine double-boiler style, placing the mixing bowl I was using over simmering water, thereby creating one less dirty dish and melting the chocolate more gently — over moderately low heat, whisking occasionally, just until melted. Remove from heat and whisk in sugar, eggs, vanilla, and a pinch of salt until well combined. Whisk in flour until just combined and spread in baking pan.
Make cheesecake batter: Whisk together cheesecake batter ingredients in a small bowl until smooth. Dollop over brownie batter, then swirl in with a knife or spatula. [I actually like using a butter knife because the tip of it is round enough that you can use it to fold bits of the brownie batter over the cheesecake batter for a more visibly marbled effect. Try it!].
Sprinkle chocolate chips over cheesecake/brownie batter swirl. Although it might be more logical to just mix them into the cheesecake batter (and feel free to do this, it will have no ill-effect) I thought it might interrupt the swirl-ability of it.
Bake brownies: Bake until edges are slightly puffed and center is just set, about 35 minutes. Serve warm or at room temperature.