Adjust oven rack to lower-middle position and preheat oven to 350 degrees. To the bottom of a 9-inch round by 2-inch tall cake pan (see note 1), add 4 tablespoons butter.
Place pan in preheated oven and heat butter until melted, about 2 to 4 minutes. Brush butter up sides of pan.
Sprinkle brown sugar over the melted butter, then stir to combine. Pat brown sugar mixture into an even layer.
Arrange pineapple slices in a single layer over brown sugar mixture. Add one cherry to the center of each pineapple ring. Add more cherries if desired (for decoration).
In a medium bowl, whisk together flour, baking powder, and salt. In an electric mixer on medium speed, or by hand using an electric mixer, cream together remaining 8 tablespoons of butter and granulated sugar until fluffy and light, about 3 to 6 minutes.
One at a time, beat in the eggs and egg white. Beat in the vanilla. Reduce speed to low.
Add 1/3 of the flour mixture and beat until incorporated. Add 1/2 milk and beat until incorporated. Repeat with 1/3 of dry mixture, then milk. Add remaining dry mixture and mix until incorporated (batter will be thick).
Pour batter over the pineapple and smooth top. Gently tap the pan on counter to remove any air bubbles and settle the batter.
Bake until a toothpick inserted into the center comes out with few, moist crumbs attached, about 45 to 50 minutes. Rotate pan halfway through to ensure even baking.
Allow cake to cool in pan 10 minutes. Place a large plate or serving platter on top of cake pan, turn plate and pan over (careful; the pan will still be warm). Allow cake to release itself onto the platter, about 1 minute, without shaking or tapping the pan. Remove pan and allow cake to cool completely, about 2 hours.