• Preheat the oven to 350ºF. Mist the sides of a 9-inch round cake pan with cooking spray.

To make the caramel: 

• In a small pot, combine the brown sugar, butter, and honey and heat over medium heat. Do not stir. Gently shake the pan to help melt the butter and the sugar. Once the mixture turns to liquid, cook until it starts to bubble and develop a nice amber color, 3 to 4 minutes, then immediately remove from the heat. Carefully pour into the prepared cake pan (it’s very hot!) and set aside.

To make the cake: 

• Put the apples cut-side down on a cutting board and slice straight down into 1⁄4-inch-thick half-moons. Starting at the outer edge of the pan and working your way toward the center, layer the apple slices on top of the caramel in a pinwheel shape, with each slice overlapping just a bit. Cut any leftover slices into small cubes to add to the batter later. Finely zest the yuzu over the apples. Squeeze 1 tablespoon of juice from the yuzu, sprinkle over the apples, and set the pan aside.
• In a small bowl, mix the flour, baking powder, and salt until well combined. Set aside.
• In a stand mixer fitted with the whisk, beat the eggs on high speed until very frothy, 3 to 4 minutes. Add the granulated sugar and continue mixing on high speed until the mixture looks shiny, another 5 minutes. Add the rum, vanilla, and cinnamon and mix on low speed for a few seconds, just to incorporate. Remove the bowl from the machine.
• Using a rubber spatula, fold in one-third of the flour mixture until just incorporated. Add half of the melted butter and fold to incorporate. Fold in another one-third of the flour mixture until just incorporated. Add the remaining melted butter, folding just a few times. Fold in the remaining flour mixture and the cubed apples until just incorporated. Fold a few more times, then pour the batter into the pan and spread in an even layer. 
• Bake until a cake tester inserted in the center comes out clean, 50 minutes to 1 hour. Transfer to a cooling rack to cool in the pan until it is no longer hot, but still warm, 30 minutes to 1 hour. Slide a butter knife around where the cake meets the pan to help loosen the edges. Flip the cake over onto a platter and give it a tap to unmold. Slice and serve immediately with vanilla ice cream, if desired. Stored in an airtight container, the cake can keep for up to 1 week in the refrigerator.