First, we are going to start by making the blueberry compote. Place the blueberries and the sugar into a saucepan over low heat and let it cook for about 10 minutes. This mixture will begin to thicken. Once thickened, mash the blueberries and remove from the heat. Allow this mixture to completely cool.

The second step is to make the brownie layer. Begin by melting the butter and dark chocolate together in a double boiler or microwave.

Once the chocolate is completely melted, add in the cocoa powder, powdered sugar, and salt. Stir until it is well combined. 

Let the mixture cool down for 5 minutes and then mix in one egg at a time, whisking between each addition. 

Add in the flour next and stir it until everything is well combined. Depending on the flour, the mixture at this stage can feel very thick. Add the water and mix until a smooth batter forms. Reserve around 3 tablespoon of the batter, this is to mix in with the cheesecake layer.

Pour the rest of the brownie batter into a greased or parchment paper lined brownie pan and spread smoothly. 

The last step is the cheesecake layer. Place the cream cheese, sugar, and vanilla extract in a bowl and mix it until smooth.

Add the egg and give it another stirring until it is evenly mixed. 

Pour the cheesecake mixture over the brownie mixture and spread it out evenly. 

To create the swirls, take half tablespoon portions of the reserved brownie and place them randomly on top of the cheesecake layer. Take one teaspoon portions of the cooked blueberry compote and place it in dollops in the remaining space on the cheesecake layer. Now take a toothpick and start to run it through the cheesecake layer in horizontal and vertical lines to create the swirls. Run the toothpick a couple of times until you are happy with the pattern.

Bake the blueberry cheesecake brownies in a preheated oven at 350 degrees Fahrenheit for around 40 minutes or until the top of the cheesecake turns slightly golden.