Step 1

Arrange 2 racks to divide the oven into thirds and heat the oven to 350°F. Line 2 baking sheets with parchment paper.

Step 2

Heat a medium skillet over medium heat. Add 1 cup chopped walnuts and toast, stirring frequently, until golden-brown and fragrant, 3 to 5 minutes. Transfer to a plate to cool.

Step 3

Cut 1 stick unsalted butter into small cubes and add to the skillet. Melt over medium heat, then continue to cook until the butter is browned and nutty smelling, 3 to 5 minutes. Carefully pour into a small heatproof bowl, making sure to scrape in any brown bits from the bottom of the pan. Set aside to cool slightly.

Step 4

Meanwhile, place 1 cup all-purpose flour, 1 cup whole wheat flour, 1 teaspoon kosher salt, 3/4 teaspoon ground cinnamon, 1/2 teaspoon baking soda, 1/4 teaspoon freshly ground nutmeg in a medium bowl and whisk to combine.

Step 5

Mash 1 medium banana in a largebowl. Add the browned butter and cold 1 large egg and whisk to combine. Add 1/2 cup packed brown sugar, 1/2 cup granulated sugar, and 2 teaspoons vanilla extract. Mix with a spatula or wooden spoon until combined.

Step 6

Add the flour mixture and mix firmly until mostly incorporated. Add the walnuts and mix until no dry streaks remain. Let rest for 5 minutes.

Step 7

Scoop the dough out in 2-tablespoon portions onto the baking sheets, spacing them about 2 inches apart, 12 per sheet.

Step 8

Bake for 8 minutes. Rotate the baking sheets from front to back and between racks, and bake until the bottoms are golden brown, 5 to 8 minutes more. Let the cookies cool on the baking sheets for 5 minutes, then transfer to a wire rack and let cool completely.