Preheat oven to 350°F (180°C). Line a 9-inch square baking pan with parchment paper, letting excess extend over sides of pan.

In a medium saucepan, melt butter over medium heat. Cook until butter turns a medium-brown color and has a nutty aroma, about 10 minutes. Remove from heat.

In a large bowl, whisk together flour, baking powder, and salt. Whisk in browned butter and vanilla. (Mixture will be thick.)

In the bowl of a stand mixer fitted with the whisk attachment, beat brown sugar and eggs at high speed until tripled in volume, 4 to 5 minutes. Add one-third of sugar mixture to flour mixture, stirring just until combined. Fold in remaining sugar mixture in two additions until well combined. Spread batter into prepared pan. Arrange Rum-Roasted Pineapple, cut side down, on top of batter, slightly pressing into batter.

Bake until a wooden pick inserted in center comes out clean, 30 to 40 minutes, covering with foil after 20 minutes of baking to prevent excess browning. Let cool completely in pan on a wire rack, at least 2 hours. Serve with warm Brown Sugar-Ginger Glaze.