Special equipment: One 3-inch deep 9-inch springform pan; a 10 by 15-inch jellyroll pan
Position a rack in the middle of the oven and preheat to 350 degrees F. Butter the bottom of the springform pan and line with parchment or waxed paper. Set aside.
To make the base: Beat together the butter and sugar by hand until light and fluffy. Beat in the yolk until smooth. Combine the flour, baking powder, and salt. With a rubber spatula, gently fold into the butter mixture. The mixture will be crumbly.
Place the dough in the pan and use your hands to pat it down evenly and firmly over the bottom. Bake for about 25 minutes, until the crust is golden and baked through. Transfer to a rack and reduce the oven temperature to 325 degrees F.
To make the batter: In the bowl of a heavy-duty mixer fitted with the paddle attachment, beat the cream cheese on the lowest speed just until smooth, no more than 30 seconds. Stop the mixer and scrape down the bowl and beater. Add the sugar in a stream, mixing for no more than 30 seconds. Stop and scrape again. Add 1 cup of sour cream and mix only until it is absorbed, no more than 30 seconds. Repeat with the remaining sour cream. Add the eggs 1 at a time, mixing only until each is absorbed; stop and scrape after each addition. Beat in the vanilla extract.
Wrap heavy-duty aluminum foil around the bottom of the springform pan so it comes at least 1 inch up the sides. Pour the batter into the pan. Place the pan in a jellyroll pan or roasting pan and pour warm water into the pan to a depth of 1/2-inch.
Bake the cheesecake for about 55 minutes, or until it is lightly colored and firm except for the very center. Remove from the oven and lift the cheesecake out of the hot water. Remove the foil and let cool completely on a rack. Wrap the cheesecake and chill overnight.
To unmold the cheesecake, run a knife or thin spatula around the inside of the pan pressing the knife against the pan, not the cake. Unbuckle the pan side and lift off. Leave the cake on the base, or run a spatula under the cake base and slide the cake onto a platter.