Preheat the oven to 350°F. Lightly butter a 9-inch glass pie plate.
Cut pineapple quarters crosswise into 1/2-inch-thick wedges. Put pineapple in shallow bowl or baking dish. In a small bowl, whisk together rum and vanilla and pour over pineapple. Let sit 5 minutes.
Heat a large skillet over medium-high heat. Using tongs, transfer pineapple to pan in batches, and cook, using tongs to turn pineapple constantly, until browned on each side, 5 to 10 minutes.
Cover bottom of pie plate with 1 layer of pineapple. Cut remaining larger pieces to fit in between spaces wherever possible. Pour juices from dish pineapple marinated in over pineapple.
In a medium bowl, cream together butter and sugar until light and fluffy. Add eggs and beat until smooth.
In a small bowl, whisk flour and baking powder together. Add flour mixture to butter mixture and stir until smooth. Scrape batter over pineapple and smooth with spatula. Bake for 30 to 40 minutes, or until a knife inserted in center of cake comes out clean. Let cool in the pan on a rack for 15 minutes, then un-mold onto a plate. Serve warm or cover and refrigerate up to 3 days.