Preheat oven to 375 degrees. Line baking sheets with parchment paper.
Whisk together flours, salt, and baking soda in a small bowl; set aside.
Put butter and sugars into the bowl of a stand mixer fitted with the paddle attachment; mix on medium speed until pale and fluffy. Reduce speed to low. Add egg and vanilla; mix until combined. Mix in banana.
Add flour mixture; mix until just combined. Stir in oats, chocolate chips, and walnuts.
U|sing a 1 and 1/2-inch ice cream scoop, drop dough onto baking sheets, spacing about 2 inches apart.
Bake cookies until golden brown and just set, about 12 to 13 minutes.
Let cool on sheets for 5 minutes. Transfer cookies to wire racks; let cool completely.
Cookies can be stored in airtight containers up to 2 days.