Using a fork or potato masher, mash the bananas until smooth, and you have enough to measure one cup. Transfer the banana to a small saucepan and bring to a simmer over medium heat, stirring often to prevent sticking, until reduced to about 1/2 cup, about 8 minutes.
The puree will thicken and darken. Remove the pan from heat and let cool. (It's hot, so resist tasting!)
Meanwhile, preheat the oven to 375°F. Line a baking sheet or two with parchment paper and set aside.
In a large bowl using an electric mixer, beat the butter and brown sugar on medium speed until pale and creamy, about 2 minutes.
Reduce the speed to medium-low and beat in the cooled, reduced banana, the egg and vanilla just until blended. Add the flour, cinnamon, baking powder, baking soda, and salt and beat just until blended, stopping to scrape the bowl as needed. Fold in the nuts.
Scoop the dough by rounded tablespoons, roll each into a ball and place about 2 inches apart onto the cookie sheet. Flatten the cookies slightly with your hand.
Bake until the tops are set and a toothpick can be inserted in one and comes out clean, 9 to 11 minutes. Let the cookies cool on the baking pan for 10 minutes, then transfer them to a cooling rack to cool completely.
Make the icing: In a medium bowl and using an electric mixer, beat the cream cheese, butter, and vanilla on medium speed until light and fluffy, 2 to 3 minutes, scraping down sides of bowl as needed.
With the mixer on low speed, beat in the powdered sugar until creamy. Spread the icing over the cooled cookies and garnish with finely chopped nuts, if desired.
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