Nov 1, - Sure, you can buy smoked chicken breasts in the supermarket but As soon as it boils, turn off the heat and add 2 cups really cold water. I fired up the pit and got my Big Green Egg (with plate setter) steady at F (C). the Big Green Egg makes awesome smoked chicken! Remove each chicken from the brine and rinse under cold running water. Take the chicken out of the fridge and insert a temperature probe into one breast of each chicken.