Step 1: Make the Blueberry Sauce

1. In a small saucepan, combine the blueberries, sugar, cornstarch, and water.
2. Cook over medium heat, stirring frequently, until the blueberries burst and the sauce thickens, about 5-7 minutes.
3. Remove from heat and stir in the vanilla extract. Let cool completely.

Step 2: Prepare the Cheesecake Swirl

1. In a mixing bowl, beat the cream cheese and sugar together until smooth.
2. Add the egg yolk and vanilla extract, mixing until fully combined. Set aside.

Step 3: Make the Cake Batter

1. Preheat your oven to 350°F (175°C) and grease your cake pan.
2. In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.
3. In a large bowl, cream the butter and sugar together until light and fluffy.
4. Beat in the eggs, one at a time, followed by the vanilla extract.
5. Mix in the sour cream and milk until combined.
6. Gradually add the dry ingredients to the wet ingredients, mixing just until no streaks of flour remain.

Step 4: Assemble the Cake

1. Pour half of the cake batter into the prepared pan and spread evenly.
2. Dollop half of the cheesecake mixture and half of the blueberry sauce over the batter. Use a skewer or knife to gently swirl them together.
3. Repeat with the remaining batter, cheesecake mixture, and blueberry sauce, swirling the top layer for a beautiful design.

Step 5: Bake the Cake

1. Bake for 40-50 minutes, or until a toothpick inserted in the center comes out clean.
2. Let the cake cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.