1. In a large mixing bowl, whisk together the eggs, brown sugar, flour and cinnamon. Set aside.
2. In a blender, puree the bananas. Set aside.
3. In a heavy pot over medium heat, add 3 cups of the milk and bring to a simmer, stirring frequently to avoid scorching. Just as the milk begins to boil remove from the heat and whisk into the egg mixture in a slow, steady stream (this helps avoid lumps).
4. Return the milk and egg mixture to the medium heat in the heavy pot and bring to a gentle simmer, stirring constantly to avoid scorching. Stir in the vanilla flavoring, then the banana puree and pour the mixture into a clean bowl.
5. Chill for 2 hours and serve.
For a decorative twist, layer into parfait glasses, alternating pudding and ginger snap cookies, then refrigerate 2 hours, and serve garnished with whipped cream and a mint sprig.