In a large bowl, mash bananas into a fine (slightly chunky) pulp. In a separate bowl, whisk together flour, baking powder and salt. Set aside.
Add 1 1/3 cups sugar, oil, molasses and vanilla to the bowl with bananas. Beat with an electric mixer on medium speed until smooth, about 2 minutes. Gradually add flour mixture and beat until just blended.
Cover bowl with plastic wrap and refrigerate dough for at least 45 minutes, or up to 1 day.
When ready to bake, preheat oven to 350 degrees F. Line two baking sheets with parchment paper.
In a small bowl, mix 1/4 cup sugar and cinnamon. Shape cookie dough into 1-inch balls (about 1 tablespoon of dough). Roll each ball into the cinnamon-sugar mixture, then place onto baking sheets about 2 inches apart. Gently flatten each cookie into a thick disk.
Bake for 8-10 minutes, until tops are crinkly and slightly puffed. Transfer to a wire rack to cool completely.