Use the paddle attachment on a stand mixer and beat 3 blocks of cream cheese until they are smooth and creamy.
Add in 1 cup of white sugar and 1 cup of sour cream. Then, beat the mixture again for a couple of minutes until thoroughly combined. Scrape the sides of the bowl with a rubber spatula to be sure that all of the ingredients are getting mixed together.
With the mixer still running, add 3 eggs one at a time until they are thoroughly mixed in. Add 1 tablespoon of clear imitation vanilla extract with the last egg.
Fold the 1/2 cups of sprinkles into the cream cheese mixture and pour it over the cooled Oreo cracker crust.
Wrap the bottom of the cake or springform pan well with aluminum foil so that it comes up several inches around the bottom of the pan. This is to prevent water from getting into the cheesecake when you bake it in a water bath.
Place the foil-wrapped cheesecake pan in a large baking or roasting dish.
Put the dish with the cheesecake in it on the center rack of the oven heated to 350°F. And then, pour a few cups of water into the baking dish until there are between 1-2 inches of water up the side of the cheesecake pan.
Bake the birthday cake cheesecake for 50 minutes to an hour. To check if it is done, shake the pan very gently. The cheesecake should be mostly firm with a slight jiggle in the center, and the top should look dry. There may be a few browned spots, but you don’t want the top to be burnt.
Turn off the oven and leave the door open, leaving the cheesecake in for about an hour so that it cools slowly.
Remove it from the oven after the hour is up and let it sit at room temperature for another hour before covering it well and refrigerating the cheesecake overnight. It won’t be completely set up and solid until it has been chilled, so try not to cut into it too soon!