Add all the ingredients for the brine to a large stock pot and heat over medium-high heat, stirring occasionally, until the salt is completely dissolved.
Remove the brine from the heat source and let it come to room temperature then place it in the refrigerator overnight or until completely chilled.
Once the brine is ready, transfer it to a food grade container that's large enough to hold both the meat and the brine.
Plunge the turkey breasts side by side in the cold brine, making sure that they are not stacked, and then place a heavy object such as a plate over them to insure that they stay submerged for the entire curing time, 4 to 5 days.
Transfer this whole rig to the fridge and check on it daily to make sure that the meat is completely submerged at all times.
After the turkey breasts are done curing, rinse them thoroughly under cold running water and place them in a large stockpot; fill with enough water to cover the meat completely. Bring to a boil then kill the heat and leave the breasts in the hot water for 15 to 20 minutes, which which will help raise the internal temperature of the meat and draw some of the salt out.
Meanwhile soak about 4 cups of wood chips in water for 10 to 15 minutes and preheat your outdoor grill to 275°F. Turning on a single burner to low should do the trick.
Make 4 wood chip pouches out of aluminium foil. For each pouch, cut out a 12” x 24” piece of heavy duty aluminum foil (double that up if using lighter foil) and place about a cup of wet wood chips on one end of the foil. Add a handful of dry chips to that, then fold the foil over the wood chips. Fold all four edges toward the center at least twice, then poke holes on the top and bottom of the pouch with a fork.
Lift the grill that’s above the lit element and place a pouch directly on the heat source. Close the lid and wait until smoke starts to come out of the pouch.
Take your turkey breasts out of their hot water bath and place them on the unlit side of the grill; close the lid.
Smoke the breasts for a total of 2 to 2½ hours, replacing the wood chips pouch with a fresh one every 30 minutes or so. If necessary, crank up the heat under the new pouch until smoke starts to come out then bring the heat back down to low.
Try and keep the heat inside your grill as stable as possible, at around 275°F. Note that it’s not necessary to get huge amounts of smoke in order to get good flavor from it. However, if you feel you are not getting enough, feel free to add more dry chips to your foil pouches, or place an aluminum container with a handful of dry chips next to your smoldering foil pouch.
Wrap your turkey breasts in aluminium foil for the final 45 minutes of cooking; your turkey pastrami will be ready when the internal temperature of the meat registers 170°F
Allow the smoked turkey to cool at room temperature for about 30 minutes then refrigerate until completely chilled; slice thinly and serve.
Your turkey pastrami will keep for up to a month in the refrigerator.