Begin by preheating a large outdoor grill. Using a paper towel folded in 4, wipe down the grill with olive oil to clean it and also create a nonstick surface.
Prepare the compound butter by roughly chopping 1/2 the basil leaves and adding it along with the butter and salt and pepper, to taste, to a food processor. Pulse until fully combined and creamy.
Using a chef's knife, make slits in each side of the fish, place a few slices of lemon and basil in each of the slits. Stuff the remaining herbs and lemon inside of the fish and head. Paint the whole fish with the basil butter and refrigerate the remaining butter to serve with the finished fish. Carefully place the fish on the grill and grill for 10 to 12 minutes until it lifts off easily - use 2 large spatulas. Turn the fish over and grill for 10 minutes more.
Serve on a large platter family style and dot with the remaining basil butter. Sprinkle with sea salt and fresh basil leaves. Serve with lemon wedges.