Preheat oven at 350 degrees. Prepare 12 cup bundt pan with non stick baking spray or grease and flour pan.
Beat cream cheese in small bowl with electric mixer on high speed until fluffy and smooth. Beat in granulated sugar and 2 egg yolks, 1/8 tsp of peppermint extract, and 3 drops of green food coloring until smooth, set aside.
Beat cake mix, 1 1/3 cups water, oil, 2 whole eggs and 2 egg whites, in large bowl on low speed about 30 seconds, then on medium speed for 2 minutes.
Pour half the batter into prepared bundt pan (about 2 1/2 cups). Spoon cream cheese mixture into center of cake batter in pan. Spoon remaining cake batter in bowl over cream cheese mixture.
Bake 45 to 50 minutes, or until toothpick inserted in cake comes out clean. Cool in pan about 20 minutes. Turn upside down onto wire rack, remove from pan. Cool completely, about 35 minutes.
Heat chocolate chips and shortening until melted, set aside.Mix powdered sugar and 1/4 tsp peppermint extract , 3 drops green food coloring, the corn syrup and enough water, 2 to2 1/2 tsp to make a thick glaze that can be easily drizzled. Drizzle powdered sugar glaze over cake. Immediately spoon melted chocolate over glaze in 1/2 inch wide ring. Working quickly pull toothpick through chocolate to make swirls. Refrigerate until serving, Store coveredin refrigerator.