Set a rack in the middle of the oven and preheat to 350°Line a 9-inch round cake pan with parchment paper, then grease the bottom and sides with butter and sprinkle with flour, shaking off any excess flour.
In the bowl of a mixer set with the paddle attachment, beat the butter on medium speed until soft and creamy, a couple minutes. Incorporate the sugar and cinnamon and continue beating until well mixed and fluffy, a few minutes more. Add the egg yolks and pureed chickpeas and continue beating.
In a small bowl, combine the flour, baking powder, and salt. Add into the mixer and continue beating until well incorporated and you have a homogenous batter.
In another bowl, whisk the egg whites until they hold stiff peaks, or do it in the mixer in a clean bowl using the whisk attachment. Fold the egg whites into the chickpea batter gently, until completely incorporated.
Pour into the prepared cake pan and bake for 30 to 35 minutes, until a toothpick inserted into the middle comes out moist but not wet. Cool for at least 10 minutes before removing from the pan. Invert from the cake pan onto a plate, then invert from the plate onto a platter so the cake is right side up.