Prepare a grill for medium-high heat cooking. Clean the grates once hot.

Cut 3 deep slits crosswise on each side of fish.

Rub all over with 2 1/2 tablespoons of the oil, sprinkle all over (including inside belly cavity) with 3/4 teaspoon of the salt and stuff the belly cavity with herbs and lemon slices.

Rub garlic bulb all over with remaining 1/2 tablespoon oil.

Grill fish until skin is blistered on the first side and not sticking to the grates, 6 to 8 minutes for branzino, 12 to 14 minutes for red snapper.

Using a large, long spatula, carefully flip fish. (If you like, use a small baking sheet as a second tool to guide the flip.)

Continue to grill until skin is blistered on the second side and flesh is opaque and easily flakes with a fork, about 5 minutes more for branzino, 10 minutes more for red snapper.

Meanwhile, place garlic, cut side down, on the grill and cook, flipping halfway through, until just charred and tender, 6 to 8 minutes.

Remove from the grill, wrap tightly in foil and let steam.

Grill lemon halves, cut side down, until just charred and juicy, about 5 minutes.

When fish is done, use 2 spatulas to carefully transfer it to a large platter and let rest for 5 minutes.

Meanwhile, squeeze cooked garlic cloves into a small bowl, discarding skins, and mash with a fork.

Squeeze grilled lemon halves into the garlic, discarding any seeds.

Stir in remaining 1/4 teaspoon salt to make a thick sauce.

Serve fish whole at the table, spooning lemon-garlic sauce over each portion.