Blueberry compote: place the blueberry and sugar in a saucepan over low heat and let it cook for about 10 minutes. This mixture should start to thicken by this time. Mash the blueberries roughly to thicken the compote. Remove it from the heat and let it cool completely.
Brownie layer: melt the butter and chocolate together in a double boiler or in a microwave. Once the chocolate is completely melted, add the cocoa powder. Stir it until it is well combined. Let the mixture cool down for 5 minutes and then mix in one egg at a time, whisking it in between each addition.
Add the flour next and stir it until everything is well combined. The mixture can feel very thick at this stage. Add the water and mix until a smooth batter forms.
Reserve around 3 tablespoons of the batter.
Pour the rest of the brownie batter into a greased and lined brownie pan and spread it smoothly.
Cheesecake layer: place the cream cheese, sugar, and vanilla extract in a bowl and mix it with an electric whisk until smooth.
Add the egg and give it another whisk until it is evenly mixed.
Pout the cheesecake mixture over the brownie mixture and spread it evenly.
Create the swirls: take half tablespoon portions of the reserved brownie and place it randomly on top of the cheesecake layer.
Take one teaspoon portion of the cooked blueberry compote and place it in dollops in the remaining space to create the swirls.
Run the skewer a couple of times until you get the pattern you want.
Bake the brownie cheesecake in a preheated oven at 350F for 40 minutes or until the top of the cheesecake turns slightly golden. Let ic cool completely and cut into slices.