Make sauce and marinate chicken:

Whisk mayonnaise, pickle brine, vinegar, Worcestershire sauce, 1 tablespoon water, garlic powder, sugar, smoked paprika, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper together in a small bowl until smooth. Transfer 1/2 cup of the sauce to a large ziplock plastic freezer bag, and add chicken. Seal bag; massage to distribute marinade. Chill 12 to 24 hours. Refrigerate remaining sauce in an airtight container for up to 4 days.

Grill chicken:

Remove chicken from refrigerator; let stand at room temperature 30 minutes. Preheat grill to high (450°F to 500°F). Remove chicken from marinade, letting excess drip off; discard marinade. Sprinkle both sides of chicken with remaining 1 teaspoon salt and 1/2 teaspoon pepper. Place chicken, skin sides down, on oiled grates; grill, uncovered, until grill marks appear and skin starts to turn golden, about 4 minutes. Flip chicken; close grill. Reduce heat to medium (350°F to 400°F), and cook, undisturbed, until a thermometer inserted into thickest portion of thigh registers 155°F, 15 to 20 minutes.

Spoon 2 tablespoons of reserved sauce into a small bowl, and brush over skin sides of chicken. Flip chicken, and continue to cook until a thermometer inserted into thickest portion of thigh registers 165°F, 2 to 4 minutes.

Remove chicken from grill, and let rest 15 minutes. Serve with reserved sauce. Suggestion: serve with Sweet Potato-and-Cabbage Slaw.