Start by pre-heating your oven to 375° Fahrenheit .
Line your springform pan (a.k.a. Cheesecake pan) with a circle of parchment on the bottom.
Cover with the 1/4 cup of softened butter, with a thicker layer at the bottom and thinner layer on the sides. I ended up not using all of the butter.
Evenly coat the sides with the 1/3 cup of sugar. Just as you did with the butter, make the bottom have a thicker layer.
Whisk together your flour, baking powder, and salt in a bowl.
With your standing mixer and the paddle attachment, cream your remaining 1 cup of sugar and 3/4 cup of butter. It should be very fluffy and a pale white.
In a separate bowl add your egg, egg yolk, and your vanilla. With the speed on low, incorporate one egg at a time. Don't forget to scrape the sides.
At a low speed, add flour mix and milk 1/3 at a time alternately until just combined.
Scrape the sides and bottom of the bowl, then add the batter to your prepared pan and even out the surface.
Place into your pre-heated oven and bake for 35-45 minutes. My cake cooked perfectly in 40 minutes.
Test the cake with a toothpick when the timer's done. If the toothpick comes out dry then the cake is done. Let it sit in the pan for 10 minutes before removing.
Now it's time to eat! Slice up and serve. Best eaten warm with some vanilla ice cream. It will surely not be a disappointment.