For the brine: Combine water and salt in a large bowl or roasting pan (you'll want something deep enough to submerge turkey in water). Place turkey breast in brine and let sit for at least 1 hour, and up to several hours.
For the herb rub: Preheat oven to 425 degrees. In a bowl, combine butter, olive oil, garlic, lemon zest, herbs, salt and pepper. Remove turkey from brine and pat dry with paper towels.
Place onions in baking dish that fits your turkey breast. Place turkey breast on top of onions. Gently separate skin from turkey breast without tearing it off (the handle end of a spatula or wooden spoon often works well.) Spread the butter-herb mixture all over the turkey, especially stuffing it under the skin and spreading it all around. If you have extra whole herbs, just place them around the turkey. If using *potatoes, drizzle them lightly with olive oil to coat and then place them surrounding the turkey. (*See note in note section)
Place turkey in preheated oven for 10 minutes and then reduce temperature to 350 without opening oven door. Continue baking until internal temperature reaches 165 degrees. Baste turkey a couple of times toward the end of baking to achieve nice golden skin. (This means use a baster, or simply a spoon, so grab juices from the bottom of the pan and pour them over the top of the turkey.) Time will vary depending on size of turkey, generally a 3-ish pound breast will take 1-1.5 hours. Use a digital thermometer and remove turkey as soon as it reaches temperature so you don't overcook.
When done, remove pan from oven and tent with foil for 5-10 minutes to let juices redistribute. Carve meat off breast bone and then slice. Make sure to pick the meat off bones- it makes great sandwiches! You may discard the onions if desired, or slice them up and serve with your meal.