In a large bowl, beat the egg yolks with white and brown sugar, rum and vanilla until light. Add melted butter, banana puree (you can mash it with a fork, but blending makes it smoother) and nutmeg. Continue beating for another minute. Then add the flour and mix it in (do not overmix).
Add the milk and beat until incorporated.
Using spatula, GENTLY fold in beaten egg whites.
Pour batter into a greased 8 inch x 8 inch (20 x 20 cm) baking dish and bake in preheated 325 F (160 C) oven for about 60 minutes or until the top is dark golden. (If after 30 minutes top is already golden, cover the cake with aluminum foil.)
Cool Banana Magic Cake in the pan for at least 3 hours before cutting.