Preheat the oven to 325°F (163°C). Generously butter a 10- to -12- cup Bundt pan. Add a tablespoon or so of flour to the pan, tilt and tap the pan to completely coat the inside with flour. Turn the pan upside down over the sink and tap to remove any excess flour. [Editor’s Note: The more thoroughly you butter and flour the pan now, the easier it will be to remove the cake from the pan.]
In a medium bowl, whisk together the eggs, rum, and vanilla extract.
In the bowl of an electric mixer fitted with the paddle attachment, on low speed, add the cake flour, both sugars, baking powder, salt, and nutmeg. Mix on low for 1 minute or until there are no longer clumps of brown sugar. Add the softened butter and mix on medium-low speed for 1 to 2 minutes.
Add the egg and rum mixture to the dry ingredients in 4 to 6 batches, mixing everything after each addition for at least 10 to 15 seconds before adding the next addition to ensure everything is well combined before adding more. Scrape the sides and bottom of the bowl with a rubber spatula. Continue this process until all of the egg-rum mixture is incorporated and the batter is smooth and no streaks remain.
Scrape the batter into the prepared Bundt pan and bake for 50 to 70 minutes. Use a toothpick to check for doneness. You want it to come out clean or with some dry crumbs clinging to it. If the toothpick has wet batter clinging to it, the cake’s not done. Let the cake cool completely in its pan on a wire rack, about 1 hour. Don't remove the cake from the pan yet.