Preheat the oven to 350°F (175°C).

In a large mixing bowl, cream together the softened butter and sugar until light and fluffy.

Add the egg and vanilla extract to the creamed mixture, and mix until well combined.

In a separate bowl, whisk together the flour, baking powder, and salt.

Gradually add the dry mixture to the wet mixture, mixing until just combined.

Press the dough evenly into the bottom of a greased 9-inch springform pan and bake for 15-18 minutes until lightly golden.

While the crust is baking, prepare the cheesecake filling by beating together the cream cheese and powdered sugar.

Add the sour cream, lemon juice, and vanilla; mix until fully incorporated and smooth.

Once the crust is done, remove it and let it cool for 10 minutes.

Pour the cheesecake filling over the cooled crust and smooth the top.

Bake the cheesecake at 325°F (163°C) for 40-45 minutes, or until the edges are set, and the center is slightly jiggly.

Turn off the oven and let the cheesecake cool in the oven with the door ajar for 1 hour.

Chill in the refrigerator for at least 4 hours, preferably overnight, before serving.