For the brownie batter: Preheat the oven to 350 degrees F. Grease an 8-by-8-inch baking dish with some nonstick baking spray. Line with a piece of parchment paper on the bottom, making sure the paper comes up over the sides a bit so it's easy to lift the brownies out when they are done.
In a bowl, mix together the flour, cocoa powder, espresso powder and salt and set aside.
In the bowl of a stand mixer fitted with a paddle attachment, cream together the sugar and butter on medium speed for 1 minute. Add the eggs and mix until combined, then mix in the melted chocolate. Add the dry mixture and mix just long enough to combine without overmixing, about 1 minute. Set aside one-quarter of the mixture and pour the rest into the prepared baking dish.
For the cream cheese filling: In a bowl using a handheld electric mixer, cream together the cream cheese, sugar, vanilla and egg very well until there are no more lumps left.
Using 2 spoons, dollop the cream cheese mixture over the top of the brownie base, then dollop the reserved brownie batter over the cream cheese mixture. Using a butter knife, swirl the brownie mixture into the cheesecake filling.
Bake the brownies until the edges are browned and the middle is just about set, about 30 minutes. Let cool for 10 minutes in the baking dish, then lift them to a cooling rack to cool completely.
