Preheat the oven to 350°F (175°C).
In a mixing bowl, whisk together the flour, baking powder, and salt.
In another bowl, cream the softened butter, granulated sugar, and brown sugar until light and fluffy.
Add the egg and vanilla extract to the butter mixture, mixing until combined.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Press the dough into the bottom of a 9-inch springform pan evenly, creating an even layer.
Bake the crust in the preheated oven for about 10-12 minutes until lightly golden.
While the crust cools, prepare the cheesecake filling. In a mixing bowl, beat the softened cream cheese until smooth.
Add the sour cream, granulated sugar, lemon juice, and vanilla extract, mixing until well combined.
Finally, add the cornstarch, mixing until just combined.
Pour the cheesecake filling over the cooled sugar cookie crust, spreading it evenly.
Bake the cheesecake for 50-60 minutes, or until the edges are set and the center is slightly jiggly.
Once done, turn off the oven and let the cheesecake cool inside with the door ajar for 1 hour.
Chill the cheesecake in the refrigerator for at least 4 hours, or overnight, before serving.