Preheat the oven to 325°F. Generously butter the bottom and sides of a heavy 9-inch diameter springform cake pan, then line it with baking parchment.
To make the banana caramel, put the sugar and 2/3 cup water in a heavy pan and cook over high heat until the sugar has dissolved. Let it boil until thickened to a golden-brown caramel, taking care not to burn it. Remove and immediately pour it into the cake pan, tipping the pan slightly from side to side until evenly coated.
Peel the bananas and halve them lengthwise. Arrange them over the caramel in a neat pattern, trimming as necessary, then dust with the ground cinnamon.
For the cake, sift together the flour, baking powder, and cinnamon into a bowl.
Put the egg whites in a separate clean bowl and whisk to stiff peaks.
Put the butter and sugar in another large bowl and whisk until light and fluffy. Slowly whisk in the egg yolks one at a time. Fold in the mashed bananas, followed by the dry ingredients. Finally, fold in the egg whites.
Pour the batter into the pan and spread evenly with a spatula. Bake for 50 to 60 minutes, or until a skewer inserted into the center comes out clean.
Remove and let cool for a few minutes before unmolding. (It is easier to unmold while it is still warm, before the caramel base hardens). Run a thin knife around the inside of the pan. Put a large flat plate over the top and, holding the pan and the plate, invert it, gently lift off the pan and peel off the baking parchment.
Serve warm.