Preheat the oven to 375 degrees F. Line baking sheets with parchment paper (or silpat), or spray with nonstick spray.
In a medium bowl, whisk together the flours, salt and baking soda.
In a large bowl, use an electric mixer to beat the butter and sugars; mix on medium speed until pale and fluffy. Reduce the speed to low and add the egg and vanilla; mix until combined. Mix in the banana. Add the flour mixture; mix just until combined. Stir in the oats, chocolate chips and walnuts.
Drop the dough by heaping tablespoonful onto the prepared baking sheets (or use a cookie scoop), spacing about 2 inches apart. Bake the cookies until golden brown and just set, 12 to 13 minutes. Let them cool on the cookie sheets for 5 minutes. Then transfer the cookies to wire racks; let cool completely.
Notes
*Be sure to use a banana that is well on its way to the trash can. The darker it is, the softer and sweeter it becomes.
*Sub milk chocolate chips for semi-sweet if you'd like.
*To toast walnuts, spread on a baking sheet and toast briefly (3-5 min) in a 375 degree oven. Watch closely because they'll turn into burnt walnuts pretty quickly.
*Cookies can be stored in airtight containers at room temperature up to 2 days. They also freeze well.