Position oven racks in top and bottom thirds of oven. Preheat oven to 350F. Line 2 baking sheets with parchment paper.
Melt butter in a medium saucepan over medium heat. Remove from heat. Whisk in sugar, orange juice, flour and salt. Then whisk in beaten egg. Stir in nuts and orange peel. Mixture will be wet.
Drop 1 tsp batter for each cookie onto prepared sheets, about 2 in. apart. Batter will spread. Bake in top and bottom thirds of oven, switching and rotating sheets halfway through, until edges are golden brown and centres are light golden, 6 to 7 min. Repeat with remaining batter.
Cool cookies on baking sheets for 2 min. Transfer to a wire rack to cool. Store in an airtight container at room temperature up to 1 week or freeze up to 1 month.
Kitchen Tip: Parchment paper is best for lining baking sheets. Silicone non-stick mats make cookies less crispy.