This recipe originally called for enveloping the fish in fig leaves. They permeate the flesh, giving it a nutty flavor, and serve as both a shield to protect the flesh and a beautiful platter. But we’ve simplified things so you can grill the fish in a foil packet. Don’t fret if you can’t find turbot is not in season. Look out for other flatfish such as Dover sole or flounder.
Prepare a grill for medium-low indirect heat (for a gas grill, leave one or two burners off; for a charcoal grill, bank coals on one side of grill). Tear off 4 large sheets of foil. Rub fish with 1/2 cup oil (2 Tbsp. per piece) and season with salt. Working one at a time, place a fillet in the center of a sheet of foil and top with 2 rosemary sprigs. Fold in short sides of foil over fish, then fold in long sides and roll edges together to seal.
Place pouches on cool side of grill, cover grill, and cook fish, turning once, 20–25 minutes. Open a pouch to check fish. Flesh should be slightly opaque and the tip of a knife should slide through easily. Grill a little longer if needed.
Meanwhile, combine parsley, celery leaves, and garlic in a medium bowl. Finely zest lemon into bowl, then squeeze in juice. Add red pepper flakes and mix in remaining 3/4 cup oil; season with salt and black pepper. Let salsa verde sit 10 minutes for flavors to come together.
Transfer fish to a platter and serve with salsa verde and aioli.