1. In a bowl, beat together butter and confectioners' sugar until smooth and fluffy. Mix in corn syrup, then add nuts and flour; mix to combine.
2. Wrap dough in a sheet of plastic wrap, forming it into a log 1-1/2 inches in diameter. Freeze until firm, 30 min.
3. Heat oven to 350 degrees F and line baking sheets with parchment. Unwrap log and cut into 1/4-inch-thick slices, rotating after each slice so log retains its shape. Place rounds about 3 inches apart on baking sheets. Bake until golden and lacy, 9 to 10 min. Slide cookies, still on parchment, onto countertop to cool; repeat with remaining dough. Store airtight up to 1 week. Makes 2 dozen cookies.