Preheat oven to 350°F (175°C). Grease and flour a 9-inch springform pan.

Make the crust: In a medium bowl, whisk together flour, baking powder, and salt. In a large bowl, cream together butter and sugar until light and fluffy. Beat in egg and vanilla. Gradually add dry ingredients to wet ingredients, mixing until just combined. Press dough into the bottom of the prepared pan.

Bake the crust for 12-15 minutes, or until lightly golden. Let cool completely.

Preheat oven to 325°F (165°C). Make the filling: Beat cream cheese and sugar until smooth. Beat in flour, salt, and vanilla. Add eggs one at a time, mixing well after each addition. Stir in sour cream.

Pour filling over the cooled crust. Place the springform pan in a larger roasting pan. Pour enough hot water into the roasting pan to come halfway up the sides of the springform pan.

Bake for 50-60 minutes, or until the cheesecake is set around the edges but the center is still slightly jiggly. Turn off the oven and leave the cheesecake in the oven with the door slightly ajar for 1 hour.

Remove from oven and let cool completely on a wire rack. Refrigerate for at least 4 hours, or preferably overnight, before serving.