Put the cream cheese, sugar and vanilla in a bowl and beat together until completely smooth.
Line a small baking sheet with parchment paper and spoon teaspoon sized blobs of cream cheese mixture onto the baking sheet. You should be able to make 16 blobs.
Put the baking sheet into the freezer and allow the cheesecake blobs to freeze completely before making the cookie dough (about 2 hours).
Make the Pumpkin Cookie Dough
Start by spreading the pumpkin puree out on a plate and using paper towels to blot away all excess moisture. This is important or the cookies will be too soggy. Cover the puree entirely in paper towel and let it soak up all the moisture. Repeat with fresh paper towel until hardly any moisture is coming out of the pumpkin.
Melt the butter in the microwave and then add it to a bowl with the white sugar, brown sugar and blotted pumpkin puree. Mix together until combined.
Add the egg yolk and vanilla and mix until combined.
Add the flour, baking soda, baking powder, salt, cinnamon and ginger and mix them in until a smooth dough forms and there are no more lumps of flour visible.
Refrigerate the dough for 30 minutes.
Once the dough has chilled, preheat your oven to 350°F / 180°C and line a large cookie sheet with parchment paper.
Make 1 tablespoon sized balls of dough and then flatten them slightly on the cookie sheet with your hand. Use your thumb to press a small dent in the center of the dough.
Take the cheesecake balls out of the freezer and place them in the little dents.
Take another tablespoon of dough and make another disc, then put that on top of the cheesecake balls and press the two sides of dough together to seal the cheesecake inside. Then roll them in your hands to make a smooth ball, making sure the cheesecake doesn't poke through.
Bake for 11-14 minutes, or until the cookies are slightly browned at the edges and set in the middle.
Let the cookies sit on the cookie sheet for 10 minutes, then transfer to a wire rack to cool. This is important, as if you leave them to cool on the cookie sheet they will get a soggy bottom. Let cool completely before serving.
Notes
When you press a dent in the bottom layer of cookie dough, make sure not to make it too deep. If the cookie dough is too thin on the bottom, the cheesecake may leak through. Make sure there is a thick enough base to contain the cheesecake filling. Don't skip on blotting the pumpkin puree - your cookies will be very sticky and hard to handle, and may end up soggy.