How to Make It
Rinse fish and cavity thoroughly, removing any remaining guts, until water runs clear. Pat dry with paper towels.
Gently scrub both sides of fish with coarse sea salt; rinse. Pat fish completely dry with paper towels. Place fish, top side (the gray side with eyes) up, on a wire rack set in a rimmed baking sheet. Refrigerate, uncovered, until very dry, at least 36 hours or up to 48 hours. Remove fish from refrigerator, and let stand at room temperature while heating grill.
Open bottom and top vents of a charcoal grill completely. Light a charcoal chimney starter filled with briquettes. When briquettes are flaming, pour onto bottom grate of grill; top with more briquettes, filling up to about 8 inches from top of grill. Let coals burn down completely to embers (about 1 hour). Using a shovel, spread embers in an even layer on bottom of grate. (Internal temperature should be 350°F to 375°F.)
Meanwhile, melt butter in a saucepan over medium heat. Add shallot; cook, stirring occasionally, until softened, about 5 minutes. Remove from heat; stir in 1/2 teaspoon fine sea salt, parsley, dill, thyme, lemon zest, and pepper. Let stand 5 minutes. Discard thyme before using.
Rub fish with oil, and season both sides of fish evenly with 1 teaspoon fine sea salt. Place an open, oiled grill basket on grill grates, and place fish, white side up, on one side of basket over coals. Grill, uncovered, until skin is slightly charred and pulls away easily from basket, 8 to 10 minutes, covering grill as needed to maintain temperature. Close basket around fish, flip fish, and open basket. Grill, uncovered, until fish flakes easily in the thickest part behind head, 8 to 10 minutes.
Using fish spatulas, transfer fish to a large serving platter. Using a spoon and fork, debone fish according to instructions on p. 35; discard bones. Sprinkle fillets with remaining 1/2 teaspoon fine sea salt. Serve fish with herb butter, lemon halves, and green salad.