Preheat oven to 350°F and toast coconut on a baking sheet until browned and smelling lovely (about 10 minutes).
Whisk flour, cornstarch, baking powder, baking soda, and salt in a small bowl and set aside.
Combine toasted coconut, sugar, and lime zest in a food processor and pulse for 30 seconds until combined. Transfer sugar mixture to mixer bowl and beat with butter until creamy. Beat in egg and vanilla until combined. Reduce speed to low and gradually add flour mixture, beat just until just incorporated.
Scrape dough into 8 inch square pan lined with parchment paper and coated with nonstick spray. Press dough evenly into pan. Bake bars for 20 minutes, until edges are golden and a toothpick inserted in the center comes out clean. Cool completely in baking pan while making frosting.
For the frosting, beat butter and salt on medium high for 30 seconds until creamy. Reduce speed to low and gradually add powdered sugar. Beat in lime zest, juice, vanilla, and milk until smooth. Increase speed to medium-high and beat until light and airy.
Lift parchment and place cookie bars on a cutting board. Use an offset spatula to spread frosting over bars, and sprinkle with lime zest and coconut chips. Chill for 15 minutes in the refrigerator to set frosting before cutting into squares with a sharp knife.