If you're using self-basting turkey breast halves (such as frozen Butterball) or kosher turkey breast halves, do not brine in step 1, but season with salt in step 2. Often, boneless turkey breast halves are sold in elastic netting; be sure to remove the netting before brining or cooking.
1. Dissolve 6 tablespoons salt in 3 quarts cold water in large container. Submerge turkey in brine, cover, and refrigerate for at least 30 minutes or up to 1 hour.
2. Adjust oven rack to middle position and heat oven to 325 degrees. Remove turkey from brine and pat dry with paper towels. Tuck tapered end of each breast underneath and loosely tie lengthwise with kitchen twine. Tie breasts crosswise at 1 1/2-inch intervals to make tidy, even roasts. Season turkey with pepper.
3. Heat 12-inch cast-iron skillet over medium heat for 5 minutes. Add oil and heat until just smoking. Brown turkey on all sides, about 10 minutes. Flip turkey skin side down, transfer skillet to oven, and roast until turkey registers 160 degrees, about 1 hour, flipping breasts halfway through roasting.
4. Using potholders, remove skillet from oven. Transfer turkey to carving board, tent loosely with aluminum foil, and let rest while making chutney.
5. Being careful of hot skillet handle, pour off all but 1 tablespoon fat. Add shallot, ginger, mustard seeds, and 1/2 teaspoon salt and cook over medium heat until shallot is softened, about 30 seconds. Stir in cranberries, sugar, water, and orange zest. Bring to simmer and cook, stirring occasionally, until cranberries have mostly broken down and mixture is thickened, about 15 minutes. Off heat, stir in vinegar and any accumulated turkey juices. Remove twine from turkey and slice into 1/4-inch-thick slices. Serve with chutney.