I had a half turkey breast (bone in) sitting in the freezer and decided to toss it on the grill.
The recipe I selected had a brine and a rub that had an intriguing mix of sweet and savoury.
The result was an incredibly moist turkey breast. The rub added a great hit of flavour - which is something that turkey needs. Yes, I am firmly in 'that' camp..
Brining a turkey breast is a great way to ensure a moist finished product but also a super way to add in some flavour. This is a fairly standard turkey brine of 2 quarts of water and ¼ cup of salt with some added flavors with the garlic, brown sugar, and bay leaf. You can use the brine to introduce other favours that you want. Add some apple cider, perhaps a quartered orange. Go wild. But the most important thing is the salt to fluid ratio stay approximately the same. It is not an exact science, so just keep it close.
I don't always remember to brine poultry and yet when i do I am amazed at the difference it makes.
When it comes time to grill you will brown the skin some, flip the skin up and turn off half the grill and point the thicker side to the heat. You want about 350° (oven roasting temperature).
Cook until internal temp of 165° (about 1 ¾ to 2 hours). Time will vary by the weight and thickness of the turkey breast and by your grill. DO NOT COOK BY TIME. Remove from heat and let rest for 10 minutes before serving.

Grilled Turkey Breast
Brine
2 qts water
1/4 cup table salt
/4 cup brown sugar optional
2-3 cloves crushed garlic optional<
2-3 bay leaves optional
Rub
2 tablespoons brown sugar
teaspoon kosher salt only if not brining and not injected
1 teaspoon chili powder
1/2 teaspoon oregano
1/2 teaspoon cumin
2 teaspoons oil
If your turkey breast is not injected or previously brined, brining is an excellent idea. Start with a brine: 2 qt. of water and add ¼ cup each of salt and brown sugar. Crush 2-3 cloves of garlic and add 2-3 bay leaves are optional.
Trim and clean a 3 ½ to 5-pound turkey breast. Add the turkey to the brine and refrigerate for 3-6 hours.
Preheat grill to a grill surface temperature of 350° to 400°. Mix a rub of 2 tablespoons brown sugar, 1 teaspoon chili powder, ½ teaspoon each oregano, and cumin. Add 1 teaspoon kosher salt if you did not brine and if you are sure the breast was not injected.
Rinse the breast carefully, pat dry then brush with a little oil. Apply the rub.
Grill skin-side down for about 5-8 minutes to get some browning of the skin then flip and turn off the direct heat on that side of the grill with the meat. Place the thick side of the breast towards the heat. Keep the temp of the grill 350° to 375° in the area between the direct and indirect sides. Keep the lid closed and your hands off as much as possible.
Cook until an internal temp of 165° in the thickest part and several other locations. This will vary by the weight and thickness of the turkey breast and your grill. Generally about 1 ¾ to 2 hours). DO NOT COOK BY TIME ALONE. Remove from heat and let rest for 10 minutes before serving.