1. To prepare basting liquid, combine first 10 ingredients in a saucepan. Bring to a simmer over medium heat. Remove from heat; refrigerate in an airtight container overnight or up to 2 days.
2. Combine Basic Brine and chicken thighs in a large bowl; refrigerate 3 hours. Drain; discard brine and solids. Pat chicken thighs dry with paper towels. Let stand at room temperature 30 minutes.
3. Immerse and soak the cedar planks in water for 1 hour; drain.
4. Preheat a charcoal fire to high heat. Let burn 30 minutes or until flames subside and coals are glowing. Spread coals to an even 4-inch thickness; fan excess ash from coals with a piece of cardboard.
5. Tie herb sprig bunches tightly to the handle of a long wooden spoon.
6. Arrange thighs, skin sides down, directly on the coals; grill 2 minutes, without moving. Turn, baste with basting liquid using herb brush, and grill 2 minutes. Repeat procedure twice. Remove thighs to a platter, skin sides up, and baste. Let stand 5 minutes. Add vinegar and juice to remaining basting liquid.
7. Arrange 4 thighs, skin sides up, on each plank. Baste thighs. Place planks directly on the coal bed; cover grill. Grill 11 minutes or until juices run clear when thigh is pierced with a fork. Singe herbs for 5 seconds after final basting. Finely chop herbs; reserve.
8. Combine parsley, 2 tablespoons oil, 1/2 teaspoon pepper, 1/4 teaspoon salt, and reserved chopped herbs on a cutting board. Add thighs to oil mixture; turn to coat. Sprinkle with tarragon; let stand 3 minutes. Serve thighs with dressing.
Basic Brine: Combine 5 cups water, 1/4 cup kosher salt, 1 tablespoon black peppercorns, 1 tablespoon chopped fresh thyme, 1 tablespoon sugar, 8 garlic cloves, 3 bay leaves, and 1 halved lemon in a medium saucepan; bring to a boil. Remove from heat; cool. Refrigerate brine overnight.