For the brownies: Preheat the oven to 350 degrees F. Grease an 8-inch square metal baking pan with nonstick cooking spray. Line the pan with parchment, leaving 2 inches overhanging on 2 sides. Grease the parchment with cooking spray.
In a medium saucepan over low heat, melt the butter with the chocolate, whisking until smooth and fully melted. Whisk in the cocoa powder until combined. Remove from the heat and stir in the granulated and brown sugar until combined. Whisking constantly, add the eggs and mix until fully incorporated. Add the flour and whisk until completely smooth; the mixture will be very thick. Set aside.
For the cheesecake layer: Beat the cream cheese in a bowl with an electric mixer or in the bowl of stand mixer until smooth. Add the granulated sugar and vanilla and continue to mix until fully incorporated and fluffy. Add the egg and mix until fully combined.
Spoon the brownie mixture into the bottom of the prepared baking pan, using a spatula to spread it into an even layer. Carefully pour the cheesecake mixture over the brownie layer; distribute evenly and smooth.
Stir the strawberry jam with a spoon to loosen it. Drop spoonfuls of the jam over the cheesecake mixture. Using a toothpick or skewer, swirl the jam into the cheesecake, being careful not to press too deep and disturb the brownie layer.
Fill a separate baking dish with hot water and place on the bottom rack of the oven. This will add moisture to the oven during baking so the cheesecake layer does not crack. Place the cheesecake brownie pan on the center rack above and bake for 40 minutes. Turn the oven off and leave in the oven for an additional 10 minutes.
Remove the brownies from the oven and allow to cool completely in the pan. Refrigerate the pan until the brownies are chilled, 2 to 4 hours. Using the parchment edges, transfer the brownies to a board, then remove the parchment from under. Trim off the edges and cut into 8 bars.