Preheat oven to 300F (150C), and line an 8-inch square cake pan with parchment paper, making sure to overlap the edges.
Fill a larger pan (12 inches) with water to a depth of about 1 inch and place it in the middle rack of the oven.
Set up a double boiler by filling a saucepan with water and placing a heat-proof bowl on top. The water should not touch the bottom of the bowl.
In the double boiler, combine cream cheese, oil, and milk. Whisk until the cream cheese is melted, being careful not to overheat. Remove from heat and wipe the bottom of the bowl.
In a separate bowl, whisk the whole egg. Add the yolks and whole eggs to the cheese mixture and whisk to combine.
Sift in cake flour and salt, and whisk until well combined. Set aside.
In a separate bowl, beat the egg whites on low speed until foamy. Then, add the vinegar and continue to beat until soft peaks form. Increase the speed to medium and add sugar in 3 batches, continuing to beat until a firm peak forms. Do not over-beat the meringue. Do not over-beat the meringue.
Gently fold the meringue into the cake batter in three stages using a spatula. Be sure to scrape the bottom of the bowl to ensure even mixing.
Pour the batter into the prepared cake pan, tapping it on the counter a few times to release any large air bubbles. Place the cake pan inside the larger pan in the water bath.
Bake the castella cheesecake for 1 hour 50 minutes to 2 hours, or until the cake bounces back when gently touched. You can also insert a toothpick into the center, if it comes out clean the cake is baked.
Remove the cheesecake from the oven and tap the pan on the counter a few times, this will cause the steam to release more quickly from the cake and prevent wrinkling on the surface.
Carefully lift the cake out of the pan using the parchment paper, place it on a cooling rack for 5 minutes then peel off the parchment paper. Return the cake back to the cooling and allow it to cool completely before cutting.
To serve, use a serrated knife to cut the cake to make clean slices.
The Castella cake can be stored at room temperature in an airtight container for up to 3 days or refrigerated for up to 7 days.
Enjoy your Taiwanese castella cheesecake!