Preheat oven to 375 degrees. Butter a 9-inch square baking pan; line with parchment, leaving a 2-inch overhang on two sides. Butter parchment. In a large bowl, whisk butter, brown sugar, and 1/3 cup granulated sugar. Add 1 egg; whisk until smooth. Stir in 1 cup flour, salt, nuts, and lime zest. Spread batter in pan. Bake until top is set and very light golden, 15 to 18 minutes.
Meanwhile, in a medium bowl, whisk together 2/3 cup granulated sugar, 2 eggs, and vanilla. Reserve 1/2 cup coconut; stir remaining coconut and 1 cup flour into egg mixture. Gently spread over base; top with reserved coconut. Bake until golden and a toothpick inserted in center comes out with moist crumbs attached, 25 minutes. Let cool completely on a wire rack. Using parchment, lift cake from pan and cut into 24 bars.
Store cookies in an airtight container, up to 1 week.