Beat the butter, brown sugar and granulated sugar with an electric mixer on medium speed in a large bowl until fluffy. Add the vanilla and egg and beat to combine.
Whisk together the flour, baking soda and salt in a medium bowl. Add the flour mixture to the butter mixture and beat until just combined. Stir in the butterscotch chips and walnuts.
Line 2 baking sheets with parchment paper. Scoop slightly mounded tablespoons of the dough and roll into balls. Arrange the balls about 2 inches apart on the prepared baking sheets and refrigerate for 30 minutes.
Meanwhile, position oven racks in the top and bottom thirds of the oven and preheat the oven to 350 degrees F.
Bake the cookies until they are set and golden on the bottom, 12 to 15 minutes. Let cool on the baking sheets for 10 minutes, then transfer to racks to cool completely.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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