Spread the sliced rhubarb over the bottom of an 8" cake pan, sprinkle with the sugar and dot the butter and set aside.
Sift together the flour, baking powder, and salt.
Add the egg yolks to the bowl of a stand mixer. You can also use a hand mixer if you prefer. Fit the whisk attachment to the mixer, and mix the egg yolks on low until combined.
With the mixer on medium speed, gradually add in the sugar. Increase the speed to high and beat until the mixture is light in color, 4 - 5 minutes.
With the mixer on low speed, add the hot water slowly, beating well. When you lift the whisk out of the egg yolk mixture, the mixture should all in a ribbon pattern.
Stir in the bran and vanilla then add the flour mixture and mix well.
Transfer the yolk mixtures to a large wide bowl.
Add the egg whites to the clean bowl of your electric mixer. Beat egg whites on medium-high until frothy. Increase speed to high and beat until the eggs are glossy and forms stiff peaks, about 4 minutes.
Remove 1 cup of the egg whites and vigorously fold into the batter. Gently fold in the remaining egg whites, running a rubber spatula down along the bottom of the bowl and lifting up through the center and over the top as you rotate bowl.
Pour the batter over the top of the rhubarb and bake in the center of the oven for 50 minutes or until the top of the cake is firm to the touch and a toothpick or cake tester stuck in the middle comes out clean.
Place the cake on a cooling rack, and cool for 15 minutes. Run a knife around the edge of the pan, place a plate over the pan and turn it upside-down. If the cake cools completely, it will stick to the pan.
One read asked about using a springform pan, which I advised against. The delicious caramel juices will seep out making the upside downness less fabulous and you will have a mess to clean up.