Preheat oven to 350°F. Line the bottom of a 9-inch springform pan with parchment paper. Set aside.
To make the cake, in a medium-size mixing bowl, whisk together the sugar, butter, and oil. Whisk in the egg and egg yolk until smooth. Add in the milk, sour cream, and vanilla.
⅓ cup granulated sugar, 3 tablespoons unsalted butter, 3 tablespoons vegetable oil, 1 large egg, 2½ tablespoons milk, 1 tablespoon sour cream, ½ teaspoon pure vanilla extract, 1 egg yolk
In a separate small bowl, whisk together the cake flour, baking powder, salt, and baking soda. Add into the wet mixture and stir just until the flour mixture is incorporated. Fold in the sprinkles until fully incorporated.
½ cup cake flour, ¼ teaspoon baking powder, ¼ teaspoon kosher salt, ⅛ teaspoon baking soda, ½ cup rainbow sprinkles
Pour the cake batter into the bottom of the springform pan and smooth out. Place into the oven to bake for 20 minutes or until it is browned to your liking.
Once the cake is baked, remove it from the oven and place it onto a cooling rack while you make the cheesecake batter.
To make the cheesecake batter, place the cream cheese and sugar into the bowl of a stand mixer fitted with a paddle attachment. Beat on high speed until completely smooth, about 3-4 minutes. Add in the eggs and egg yolk and beat for another minute until fully incorporated.
28 ounces cream cheese, 1 cup granulated sugar, 4 large eggs, 1 egg yolk
Add in the heavy cream, sour cream, and yogurt and beat together. Add the cornstarch, vanilla, and salt and stir until incorporated. Fold in the sprinkles. Pour the cheesecake batter on top of the cake and smooth out.
10 tablespoons heavy cream, ⅓ cup sour cream, ⅓ cup yogurt, ½ tablespoon cornstarch, 2 teaspoons pure vanilla extract, ½ teaspoon kosher salt, 1½ cups rainbow sprinkles
Wrap the bottom and sides of the springform pan with tin foil to prevent water from seeping in. Place springform pan into a larger pan to create a water bath. Fill the larger pan with enough water to come 1 inch up the side of the springform pan.
Place into the preheated oven and bake for 1½ hours or until the cheesecake is set but the center is jiggly. You may need to cover the cheesecake with aluminum foil once the cheesecake starts browning to prevent it from getting too dark.
Once the cheesecake has baked, turn the heat off and leave the cheesecake to rest in the oven for 1 hour with the door closed.
After the cheesecake has rested, remove from the oven and place onto a cooling rack to cool completely. At this point, the springform pan should be unwrapped from the aluminum foil and out of the water bath. Once the cheesecake has cooled completely, cover with aluminum foil or plastic wrap and place into the fridge to chill for at least 3 hours or overnight.
After the cheesecake has chilled, run a butter knife around the edges to release the cheesecake from the springform pan. Release the pan sides and place the cheesecake onto a cake stand or plate to slice and serve.