Preheat the oven to 350 degrees F. Prepare a baking sheet with parchment paper, a baking mat, or lightly greased.
In a large mixing bowl add the sugar, brown sugar, and butter. With an electric mix cream until well blended.
Beat the eggs into the butter and brown sugar mixture until well blended, add the mashed banana and mix well.
Combine the flour, baking soda, salt, and cinnamon. Add to the cream mixture and mix well.
Drop by round teaspoons onto prepared baking sheets.
Bake for 10 to 12 minutes or until cookies are lightly brown and spring back when lightly touched.
Remove the cookies from the cookie sheet and cool them on a cooling rack.
While the cookies are cooling prepare the frosting.
In a small saucepan add the butter, brown sugar, and cream. Over medium heat bring to a boil, stirring constantly.
Reduce the heat and continue to cook for about one minute.
Remove from the heat and stir in the vanilla.
Gradually stir in powdered sugar until you have the right consistency for frosting. Beat until smooth.
Frost the cookies when completely cooled.