Re: New Member -- Big Green EggVal, Welcome to Smoking Meat Forums. You have found the right place to learn the art of smoking and the BGE will work fine. You might be interested in looking at the Poultry forum and the charcoal forum.
To start you out, soak your turkey breasts in a brine solution that is made up of
1 gal water
1 cup brown or raw sugar
1/2 cup kosher salt (Most brine solutions call for an equal amount of sugar and salt but most turkeys and turkey product came already injected with a "Flavor Enhancer" [brine])
Add to this any additional flavorings/spices that you would like to use.
Place the turkey breasts in a non-reactive bowl (glass or plastic) and cover with the brine solution. An easier way that I use is to place a half of a turkey breast into a gallon zip-top plastic bag and then add enough brine to cover. Do this with each breast half. Place in the refrigerator for no more than 4 hours. Remove from the fridge, rinse under cold water and pat dry with paper towels. Now is a good time to rub each breast down with some yellow mustard and sprinkle on you favorite rub. Let rest for about 10 minutes or so before placing them into the BGE.
Since we're dealing with poultry here, we can hot smoke them using 325-350 degree cooking temp. If it were a tougher piece of meat like a pork butt or brisket we would go low and slow (225-250 degrees for 8+ hours or more). Add your favorite smoking wood (chunks or chips) to your pre-lit charcoal in the BGE, replace the rack and add the turkey breasts. You'll want to cook these breasts to an internal temp of 170 degrees.