Preheat the oven to 375° F.
Moisten the cake or lady fingers with the rum syrup, enough to soften, but not soaked.
Beat the eggs with the sugar, vanilla, orange flower water and a pinch of salt until they start to stiffen, about 1 minute.
Add the powdered sugar and continue beating until they double in volume, and start to stiffen, about a minute more.
Melt the butter in a 9-inch skillet over medium heat until it starts to foam without browning.
Pour in the egg mixture. Cook for about 2 minutes until, when you slip a spatula under the omelet, you can see that it has started to develop golden color on the bottom.
Put the skillet into the oven and let the omelet cook for another 3 or 4 minutes until the eggs have souffle and lightly firmed. The center should not be liquid.
Remove from the oven and arrange the cake crumbs in a line across the omelet. Allow it to sit for a minute or two before carefully inverting and rolling it onto a plate.
Add a garnish of seasonal berries, but don't be afraid to serve the omelet plain.
Slice in 4 to 6 parts, dust with powdered sugar and serve.