Preheat the oven to 350F (180C) and line a 12-cavity muffin tin with muffin papers.
Finely chop the white chocolate and place in a heatproof bowl.
Heat the milk until warm, then pour over the white chocolate. Give the mixture a stir and let sit for 2 minutes. Microwave on medium power (not high power, which is often the default) for 45 seconds and stir. If the chocolate has not quite melted yet, microwave again for 30 seconds on medium power. Set aside.
Sift together the cake flour, baking powder and salt and whisk for about 10 seconds. Set aside.
In a large bowl, beat together the butter and sugar until fluffy. Beat in the oil and vanilla extract.
Be sure that the egg whites do not feel cold to the touch, then beat them into the batter 1 at a time. Turn off the mixer and scrape down the bowl after each addition.
With the mixer on a low speed, beat about ⅓ of the flour mixture into the butter mixture. Then beat in about ½ of the milk and melted chocolate mixture.
Whisk in about ⅓ more of the flour mixture, followed by the rest of the milk. Lastly, whisk in the rest of the flour until there are no lumps in the batter. I always recommend doing the last few additions by hand to avoid overmixing the batter.
Spoon the batter into the prepared pan, distributing the batter equally between the 12 cupcake liners.
Bake in the middle of the oven for about 19-22 minutes, or until an inserted toothpick comes out clean or with a few cupcake crumbs but no batter.
Cool the cupcakes in the pan for at least 10 minutes, then carefully transfer to a wire cooling rack to continue cooling.