Place a rack in the middle of the oven and preheat to 350 degrees F. Cut out a circle of parchment or wax paper and place it in the bottom of a 9 x 2-inch round cake pan. Lightly butter the sides of the pan.

To make the topping: Heat a small saucepan over medium-high heat. Add the butter and cook, stirring occasionally, until it's browned and has a nutty aroma, about 5 minutes. Remove from the heat and whisk in the brown sugar and rum. Pour the butter-sugar mixture into the cake pan. Using a rubber spatula, spread the mixture out so it to covers the bottom of the pan.

Quarter the pineapple lengthwise, core, and slice it into about twenty 3/8-inch-thick slices. Arrange some of the pineapple slices in a slightly overlapping ring around the inside of the pan, leaving the center open. Place 3 of the slices in the center in a triangle pattern. Press the slices into the butter-sugar mixture.

To make the cake: Using a sieve over a large bowl, sift together the flour, baking powder, cinnamon, nutmeg, and salt. Set aside.

In a large bowl with a hand-held electric mixer, combine the butter and sugar and mix at a low speed until just incorporated. Raise the speed to high and mix until light and fluffy, about 5 minutes. (Occasionally turn the mixer off, and scrape the sides of the bowl down with a rubber spatula.)

Add the eggs one at time, waiting for each one to be fully incorporated before adding the next.

Reduce the speed of the mixer to low. Add the flour mixture in 3 additions alternating with the milk in 2 additions. Raise the speed to medium and mix briefly until a smooth batter is formed.

Pour the batter onto the pineapple-lined pan. Place the pan on a baking sheet and bake the cake, rotating the pan once during cooking, until golden brown and a toothpick inserted in the center of the cakes comes out clean, about 45 to 55 minutes. Remove the cake from the oven and let cool for 5 minutes.

Run a knife around the edge of the pan to release the cake. Carefully invert the cake onto a serving plate and remove the parchment paper. Let cool completely before serving.

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