Heat the oven to 350°F. Line two or three large baking sheets with nonstick baking liners or parchment.
In a small saucepan over medium heat, melt the butter and then continue cooking it just until the butter solids at the bottom of the pan turn deep golden brown (not black), 5 to 7 minutes. Watch the butter carefully. Immediately remove the pan from the heat and pour the butter into a small bowl, scraping the pan to get all the butter. Let cool slightly, about 5 minutes.
Combine the chopped nuts, sugar, egg, vanilla, and salt in a medium bowl. Stir until blended. Slowly add the browned butter and continue stirring until blended.
Drop the batter by slightly heaping teaspoonfuls about 3 inches apart on the prepared baking sheets. Bake until the cookies are golden brown, 6 to 8 minutes. Let the cookies cool on the sheets on racks for 5 minutes before transferring them to racks to cool completely. Repeat with the remaining batter after the sheets have cooled.
Store at room temperature or freeze in an airtight container, separating the cookie layers with waxed paper.