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Prepare cake mix as directed on package, using whole eggs. Stir in vanilla. Divide batter evenly into 3 bowls. Stir 50 drops (about 1/2 teaspoon) food color into one of the bowls, tinting batter dark green. Stir 15 drops food color into another of the bowls, tinting batter medium green. Stir 2 drops food color into last of the bowls, tinting batter light green.
Pour each bowl of batter into a greased and floured 8-inch round cake pan.
Bake as directed on package. Cool cakes on wire rack.
For the Frosting, beat butter and vanilla in large bowl until light and fluffy. Gradually add confectioners' sugar, beating well after each addition and scraping sides and bottom of bowl frequently. Add milk; beat until light and fluffy. Place dark green cake layer on serving plate. Spread with 1/3 cup of frosting. Top with medium green cake layer. Spread with 1/3 cup of frosting. Top with light green cake layer. Frost top and sides of cake with remaining frosting.