Place the chicken, skin side down, onto the Big Green Egg grill grate. Cook for 8 to 10 minutes on each side of the breast, or until a meat thermometer inserted in the thickest part of the chicken breasts registers a temperature of 170 to 180 degrees Fahrenheit.he Big Green Egg ceramic grill is designed to let you grill, smoke and bake using one product rather than three. The thick ceramic walls retain heat, allowing you to cook your meats longer at lower temperatures, much like a smoker, but opening the lid gives you the convenience of a traditional charcoal grill.