Step 1

Set the oven rack to the lower-third position. (This is right below the center of the oven.) Preheat the oven to 325°F. Line an 8-inch square pan with foil and lightly grease the foil. Set aside.

Step 2

In a medium heatproof bowl set over a pot of barely simmering water, melt the butter and chocolate, stirring constantly until the mixture is smooth. You can also microwave in 30-second intervals, stirring between each one.

Step 3

Remove from heat and whisk in the sugar until fully combined. Set aside to cool slightly, about 5-10 minutes. If the mixture is too warm, it will scramble the eggs in the next step.

Step 4

Whisk in the eggs, egg yolk, and vanilla. Continue whisking until completely smooth and shiny, about 1 minute.

Step 5

Add the flour, cocoa powder, and salt. Whisk until just combined. Once you no longer see streaks of dry flour, stop whisking. You don't want to overmix the batter.

Step 6

Reserve 1/3 cup of the batter and spread the remaining batter evenly into the prepared pan. The batter will be thick.

Step 7

In a small bowl, beat the cream cheese and sugar on until smooth. You can use a handheld mixer, whisk, or just a spoon.

Step 8

Add the egg yolk and vanilla. Mix until smooth and creamy.

Step 9

Drop spoonfuls of the cheesecake filling evenly over the brownie batter then drop spoonfuls of the reserved brownie batter in the empty spaces between the cheesecake filling.

Step 10

Use the tip of a sharp knife to gently swirl the dollops together.

Step 11

Bake for 35-40 minutes, until a toothpick inserted into the center comes out with a few moist crumbs attached. Rotate the pan halfway through baking.

Step 12

Cool the brownies in the pan for 5 minutes. Lift the brownies out by using the foil as a sling and set on a wire rack to cool completely. Refrigerate for at least 3 hours before slicing.