Preheat oven to 350. Generously butter a large bundt pan and dust with flour, knocking out the excess. Set aside.
In a medium bowl, whisk the flour, baking powder and salt. Set aside.
Peel and core the apples and cut into chunks about ¾ inch (no need to be totally precise, it’s just a guideline). Toss them with the cinnamon and ¼ cup of the sugar.
Add the eggs to a stand mixer. Beat on high speed until pale yellow, a few minutes. Add the sugar and beat another minute. Pour in the rum and vanilla and melted butter and beat again, scraping the sides to make sure it is all incorporated. Add the flour and mix until just combined.
Fold in the apples then scrape into the prepared pan. It should fill it ¾ of the way. Place in the heated oven and bake for 65-75 minutes. The sides will pull away from the pan and the cake will feel firm to the touch. Let cool completely, then turn out onto a serving plate. The cake will keep for up to 3 days, wrapped well.