For the topping and fruits: Spray an 18-by-13-inch sheet pan that is at least 1 inch tall liberally with nonstick cooking spray; set aside.
Heat a medium 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 prepared sheet pan. Using a rubber spatula, spread the mixture out to cover the bottom. Set aside to cool while you prepare the fruit.
Cut the pineapple rings into half-moons. Cut the maraschino cherries in half. Slice off both ends of the navel oranges, then cut each into 1/4-inch-thick wheels, 4 to 5 pieces total per orange. Trim the wheels into even squares, about 1 1/2 inches in size (removing the rinds in the process). Repeat with the blood oranges. Cut the plums in half and remove the pits. Cut each half into 6 wedges, for a total of 24 wedges.
With the long sides of the prepared sheet pan positioned horizontally in front of you, visualize the pan divided into 4 even sections: 2 sections in the top row and 2 sections in the bottom row. In the upper left quadrant, arrange the pineapple half-moons in rows 3 across and 5 down. Put a maraschino cherry half into each of the half-moon centers with the cut side up. You will have 1 piece each of pineapple and cherry left over, which you can save for another use or eat as a snack.
Arrange the raspberries on the upper right quadrant of the sheet pan with the holes facing down. Starting right below the raspberries in the lower right quadrant, arrange the plum wedges in rows 4 across and 6 down. Arrange the citrus pieces side by side in the lower left quadrant of the sheet pan, alternating citrus types. You might have leftover citrus pieces, which you can save for another use or eat as a snack.
For the cake: Position an oven rack in the center of the oven and preheat to 375 degrees F.
Combine the flour, baking powder, baking soda and salt in a large bowl and whisk to combine. Set aside.
Beat the granulated sugar and butter in a large bowl with an electric hand mixer on high until smooth and fluffy, about 8 minutes. Add the eggs and blend until combined, then mix in the vanilla. The mixture might look curdled, but that is okay. Reduce the mixer speed to medium. Add the flour mixture in 3 batches, alternating with the buttermilk and beginning and ending with the flour; mix until just incorporated.
Spoon large mounds of the cake batter on top of the fruit in the sheet pan. Use a large offset spatula to gently smooth out the batter. Bake, rotating the sheet pan once halfway through, until the cake is golden brown on top and springs back when you press it lightly in the center, 45 to 50 minutes. Let cool on a cooling rack for 15 minutes.
Run a paring knife or the tip of an offset spatula around the edge of the cake in the sheet pan. Put a large cutting board (it should be larger than the sheet pan) on top of the sheet pan and use 2 kitchen towels to carefully invert the sheet pan onto the cutting board. Use a large wooden spoon to tap the sheet pan all over, then gently lift away the sheet pan. Cut the cake into pieces and serve immediately while still warm.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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