In a large bowl, mix the cream cheese, cornstarch, brown sugar, and granulated sugar until smooth. Add the brown butter, sour cream, and vanilla, mixing until well combined.
Add the eggs and egg yolks one at a time, mixing until fully incorporated. Pour the batter over the prepared crust.
Place the springform pan in a larger roasting pan and fill the roasting pan with hot water until it reaches about 1 ½ inches up the sides of the springform pan.
Bake the cheesecake for 1 hour, then let it sit in the oven with the door cracked open for another hour. Once done, remove from the oven and let it cool completely before refrigerating for at least 8 hours or overnight.