Preheat oven to 300F. Grease and line a 9-inch springform pan.
In a small bowl, mix the graham crackers, ginger, cardamom, and melted butter together. The consistency should be crumbly but not too sticky. Press the graham cracker mixture into the bottom of the springform pan until it evenly covers the bottom. Set aside.
In a large bowl or stand mixer, beat the cream cheese for about a minute. Add in the sugar, ginger, and cardamom and cream together for 2-3 minutes until incorporated. With the mixer set to low, mix in the milk, vanilla, all four eggs (one egg at a time), and finally the sour cream until everything is mixed together and smooth.
Using a spatula or wooden spoon, fold in the blueberries.
Pour the batter into the springform mold and bake at 300 degrees for 1 hour.
After 1 hour has passed, turn off the oven and prop the door open using a wooden spoon (or other heat-resistant utensil) and allow to cool inside the oven until it has come to room temperature, about 3 hours. This will stop the top from cracking. The cheesecake should then be put in the fridge to cool for at least 2 hours before topping with the blueberry compote.
Meanwhile, make the blueberry topping. Place the blueberries and sugar in a sauce pan and cook over medium heat until the fruit begins to break down. This will take about 5 minutes for fresh fruit and a few minutes longer for frozen fruit.
Because frozen fruit has more water in it, you may need to drain off some of the extra liquid, otherwise, let it continue to cook down over medium heat. Make sure to stir to keep the bottom from burning.
Whisk in the cornstarch and allow the compote to cool completely.
Once the cheesecake is cool and firm, spread the blueberry compote over the top of the cheesecake. Put the cheesecake back in the fridge for at least an hour before serving.
Before serving, you can add some fresh whipped cream to the top as well, or serve as is.