Butter and flour 3 (9-inch) round cake pans; set aside. Preheat oven to 350°F.
For Cake: In a mixing bowl, stir together the flour, sugar, baking soda, cinnamon, and salt. Add the eggs and oil and stir just until the dry ingredients are moistened; do not beat. Stir in the banana, pecans, and vanilla.
Divide the batter among the prepared pans and bake for 23 to 28 minutes, or until a wooden pick inserted in the center comes out clean. Cool the cake layers in the pans for 10 minutes, and then turn them out onto a rack to cool completely.
While the cake cools, make the frosting: In a bowl, blend together the cream cheese and butter. Gradually add the confectioners' sugar and beat until light and fluffy. Beat in the vanilla. Stir in the pecans, or reserve them to sprinkle over the frosted cake.
Fill and frost the cake when it is completely cool.