Preheat oven to 375 degrees. Whisk together flours, salt, and baking soda in a small bowl; set aside. Put butter and sugars into the bowl of an electric 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 chunks, and walnuts.
Using a 1 1/2-inch ice cream scoop, drop dough onto baking sheets lined with parchment paper, spacing about 2 inches apart. Bake cookies, rotating sheets halfway through, until golden brown and just set, 12 to 13 minutes. Let cool on sheets on wire racks 5 minutes. Transfer cookies to wire racks; let cool completely. Cookies can be stored in airtight containers up to 2 days.
Use a ripe banana, which has more concentrated flavor than an unripe one.