Rinse and thoroughly dry the chicken breasts. Sprinkle all over with salt, putting some under the skin. Cover and refrigerate at least 2 hours and up to overnight. Let chicken come to room temperature 30 minutes before grilling.
Prepare a charcoal or gas grill for indirect heat: If using a gas grill, turn all burners to high and close the lid. When the temperature inside the grill reaches 400°, lift lid and turn off one of the burners. The area over the turned-off burner is the indirect heat area. If using a charcoal grill, light 50 to 60 briquettes and let burn until covered with ash, about 20 to 30 minutes. Mound them to one side. The area over the section cleared of coals is the indirect heat area.
Brush the grill with vegetable oil. Place chicken skin side down on indirect-heat area; close lid on gas grill. Cook 15 minutes. Turn chicken over, close lid on gas grill, and cook another 10 minutes. Move chicken over direct heat and cook, turning once, until skin is well browned and crisp, 3 to 5 minutes. Watch carefully; dripping fat or any added oils or butters catch fire easily (a small spray bottle filled with water is handy for taming flames).
Test one piece for doneness by cutting into the center. It should be slightly pink (it will finish cooking while it rests). If very pink, move all chicken back to indirect heat, cover grill, and cook for another 5 minutes. Let rest at least 10 minutes before serving.
Once you've got the basic method down, try adding different flavors to your grilled chicken:
Spice-rubbed. In step 1, sprinkle your favorite spice rub all over the chicken while you're letting it come to room temperature. We like the Indian flavors of 1 teaspoon ground cumin, 1 teaspoon ground coriander, 1/2 teaspoon turmeric, 1/2 teaspoon black pepper, and 1/2 teaspoon cayenne.
Marinated. In step 1, coat chicken in a zesty marinade while you're letting it come to room temperature. Most vinaigrettes make great marinades for chicken. We like to mix 3 tablespoons olive oil, 1 tablespoon fresh lemon juice, 1 teaspoon minced fresh rosemary or thyme, 1/2 teaspoon Dijon mustard, 1/4 teaspoon salt, and 1/4 teaspoon pepper.
Buttered. In step 1, while chicken is coming to room temperature, rub softened butter onto and under skin. We like to use a compound butter (softened butter mixed with seasonings, herbs, or spices). For one with a bit of kick, we blend 1/4 cup butter with 1/2 minced chipotle chile (either canned, or dried and soaked in hot water until soft), 1 teaspoon fresh lime juice, and 1/2 teaspoon salt.
Herbed. In step 1, while chicken is coming to room temperature, rub herb paste onto and under skin. Pesto (homemade or store-bought) is a good choice.
Note: Nutritional analysis is per serving.