Preheat the oven to 350°F. Lightly butter a 9-inch glass pie plate.

Cut the pineapple quarters into 1/2-inch-thick triangles. Put the pineapple in a medium shallow bowl or baking dish. In a small bowl, whisk the vanilla and rum together. Pour the vanilla and rum over the pineapple. Let sit for about 5 minutes.

Heat a large skillet over medium-high heat. Using the tongs, transfer the pineapple to the pan in batches and saute, using the tongs to turn the pineapple constantly, until browned on each side, 5 to 10 minutes.

Cover the bottom of the dish with 1 layer of the pineapple. Dice the remaining larger pieces and fit them in between the spaces wherever possible. Pour the liquid over.

In a medium bowl, cream the butter and sugar together until light and fluffy. Add the eggs and beat until smooth. In a medium bowl, stir the baking powder and flour together well. Add to the wet ingredients and stir until smooth. Scrape the batter over the pineapple and smooth with a spatula. Bake for 30 to 40 minutes, or until a knife inserted in the center of the cake comes out clean. Remove from the oven and set on a wire rack. Let rest for 15 minutes, then unmold onto a plate. Serve now, or cover and refrigerate for up to 3 days.