How to Make It
Heat oven to 350° F with a rack in the lowest position. Butter a 9-inch round cake pan. In a medium bowl, whisk together the flour, baking powder, and salt; set aside.
In a small saucepan, combine the brown sugar and 4 tablespoons of the butter. Bring to a simmer over medium heat and cook, stirring, until the mixture is slightly darkened, 3 to 5 minutes. Pour into the prepared pan.
Using a sharp knife, cut off the pineapple peel, cut the fruit into wedges, and cut out the core; thinly slice crosswise. Starting in the center of the pan, arrange the slices in concentric circles, with the rounded part of each piece toward the outside of the pan (you may not need all of the pineapple).
Using an electric mixer, beat the remaining 8 tablespoons of butter with the granulated sugar on medium-high until fluffy, 2 to 3 minutes, scraping down the sides of the bowl as necessary. Beat in the eggs, one at a time, then the vanilla. Reduce mixer speed to low. Add the flour mixture in 3 additions and the milk in 2 additions, beginning and ending with the flour mixture. Mix just until combined (do not overmix).
Carefully spread the batter over the pineapple and bake until a toothpick inserted in the center comes out clean, 50 to 55 minutes. Let cool 10 minutes, then invert cake onto a serving platter. Rearrange any pieces of pineapple that have stuck to the pan. Serve warm.