For the Salmoriglio sauce: Whisk together the lemon juice and olive oil in a medium bowl. Add garlic, capers, herbs and lemon zest. The sauce should be loose in consistency with high-acid flavor. Season to taste with salt and pepper.
For the fish: Preheat the grill for cooking over high heat (around 450 to 500 degrees F).
Bring fish to room temperature, 30 minutes, then wrap fish briefly in paper towels to dry out skin, which will help keep the fish from sticking to the grill.
Stuff the inside cavity of the fish with herbs, garlic and lemon slices, and sprinkle the cavity with salt and pepper. Brush the outside of the fish with olive oil, then sprinkle with salt and pepper. Brush a fish grill basket with oil or spray with nonstick cooking spray. Place fish in fish grill basket and secure shut. (If you don't have a fish grill basket, place the fish between 2 wire cooling racks and secure the racks together with wire.)
Place basket on the hottest part of the grill. Cook until the eye turns white and skin is charred, 6 to 8 minutes per side. Let rest 5 minutes.
Place fish on a platter and dress with the Salmoriglio sauce.