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Melt 1/2 stick butter in a medium saucepan over medium heat. Add oats, and cook, stirring, until toasted, 5 to 7 minutes. Spread oat mixture on a parchment-lined baking sheet; let cool.
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Meanwhile, preheat oven to 350 degrees. Whisk together flour, baking soda, and 1 teaspoon salt. Beat together remaining stick butter and the sugars with a mixer on medium-high speed until pale and fluffy. Add egg, and beat until combined. Add nut butter, and beat on medium speed until well combined.
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Add oat mixture and chopped nuts, and beat on low speed until combined. Add flour mixture, and beat until combined.
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Roll dough into 1 1/2-inch balls. Place cookies 1 inch apart on 2 parchment-lined baking sheets. Bake until golden, 12 to 15 minutes. Let cool completely on baking sheets.