Line a 13 x 9-inch baking pan with heavy-duty plastic wrap, allowing plastic wrap to extend 1 inch over sides of pan. Lightly coat the plastic wrap with cooking spray. Spread coconut in an even layer in bottom of pan; set aside.
Sprinkle gelatin over 1/2 cup cold water in a small bowl; set aside.
Combine remaining 1/4 cup of water, 1 3/4 cups granulated sugar, and corn syrup in a large saucepan. Cook, without stirring, over medium-high heat until a candy thermometer registers 260° (about 15 minutes). Remove from heat; gradually stir in softened gelatin (mixture will appear foamy).
While sugar mixture cooks, beat the vanilla, salt, and egg whites at high speed in a heavy-duty stand mixer with whisk attachment until foamy. Gradually add remaining 1/4 cup granulated sugar, 1 tablespoon at a time, until stiff peaks form. Gradually pour in gelatin mixture, beating until very thick (about 5 minutes). Gently spread marshmallow mixture over coconut in prepared pan. Coat 1 side of another sheet of plastic wrap with cooking spray. Place plastic wrap, coated side down, over marshmallow mixture. Chill 8 hours or until firm.
Sprinkle powdered sugar and cornstarch over a cutting board. Remove top sheet of plastic wrap. Invert marshmallow mixture over powdered sugar mixture. Using a dough scraper, cut mixture into about 1-inch squares. Store between sheets of wax or parchment paper in an airtight container.