Coat a 9-inch square metal baking pan with cooking spray; line bottom of pan with parchment paper, allowing parchment paper to extend over edge of pan.
Beat egg whites with a mixer at high speed until foamy. Gradually add 6 tablespoons granulated sugar, 1 tablespoon at a time, beating until stiff peaks form.
Place egg yolks and remaining 6 tablespoons granulated sugar in a medium bowl; beat with a mixer at medium-high speed until thick and pale (about 3 minutes). Beat in vanilla. Gently fold egg yolk mixture into egg white mixture.
Weigh or lightly spoon flour into a dry measuring cup; level with a knife.
Combine flour, baking powder, and salt; sift mixture over egg mixture, 1/3 cup at a time; fold in. Spoon batter into prepared pan, spreading evenly.
Bake at 350 for 15 to 20 minutes or until cake springs back when lightly touched. Cool in pan 5 minutes on a wire rack; remove cake from pan, using parchment paper sides as handles. Cool completely on a wire rack (with parchment paper in place). Wrap cake and parchment paper in plastic wrap; chill 2 hours.
Remove plastic wrap from cake; place cake (with parchment paper in place) on a cutting board.
Cut cake into 16 squares, using a serrated knife.
Combine milk and chopped chocolate in a small saucepan; cook over medium heat 2 minutes or until chocolate melts.
Remove from heat; stir with a whisk until smooth.
Add powdered sugar and cocoa, stirring with a whisk until smooth.
Place coconut in a medium bowl. Dip cake squares, 1 at a time, into melted chocolate mixture, using 2 forks to gently turn the cake to coat; scrape off excess chocolate on side of pan.
Place chocolate-coated cake in coconut, using 2 forks to gently turn the cake to coat.
Place lamingtons on wire rack; let stand 15 minutes or until chocolate is set.
