In a bowl, beat together butter and confectioners' sugar until smooth and fluffy, about 2 minutes on medium speed. Mix in corn syrup, then add nuts and flour; mix to combine.
Wrap dough in a sheet of plastic wrap, forming it into a log 1½ inches in diameter. Freeze until firm, 30 min.
Heat oven to 350°F and line baking sheets with parchment.
Unwrap log and quickly cut into ¼-inch-thick slices, rolling 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.Chocolate Sandwich Variation: Once cooled, you can make chocolate sandwich cookies by spreading a little melted chocolate on the bottom of one cookie and sandwiching it with another. Let cool and set and serve.