Generously butter a 9-inch (23 cm) round cake pan, making sure to coat the bottom and sides.

In a large saucepan, add the sugar and melt over medium heat without stirring. When the caramel starts to form, gently stir until you get an amber-colored caramel. Immediately pour the caramel into the prepared pan (work quickly, as it hardens fast). Make sure the entire bottom is coated with caramel.

Let the caramel cool and harden. Meanwhile, melt the butter in a small saucepan and set aside.

Peel, core, and slice the pears into 8 wedges each. Arrange the slices in a rosette pattern in the pan (if you have extra slices, layer them in the center).

Preheat the oven to 350°F (180°C).

In a stand mixer or with an electric mixer, beat the 3/4 cup sugar and whole eggs on high speed for about 3 minutes, until the mixture is pale and fluffy. With the mixer running, slowly pour in the cooled melted butter and continue beating on high for about 1 minute. Reduce the speed to low and add the flour, baking powder, and rum. Then increase the speed to high for 1 minute, until the batter is smooth and creamy.

Spread the batter evenly over the pears.

Bake for 45 minutes at 350°F (180°C). Keep an eye on the cake as it bakes.

Remove the pan from the oven and place it on a wire rack. Let it cool for just 30 seconds to 1 minute (no longer, or the caramel will harden!). Run a thin knife around the edge to loosen the cake, then invert the pan onto a serving plate to unmold.

Let cool for 15 minutes. Enjoy warm or store under a cake dome to keep the cake soft and the pears tender.