Pat the fish completely dry with paper towels. Using a sharp knife, make three diagonal slashes on each side, cutting down to the bone. Rub 1 tablespoon olive oil all over both fish, then season generously inside and out with salt and pepper. Stuff the cavities with lemon slices.

In a food processor, pulse the parsley, mint, capers, garlic, and anchovies until finely chopped but not pureed. Transfer to a bowl and stir in the vinegar, extra virgin olive oil, and lemon juice. Season with salt to taste. Let sit at room temperature while you grill the fish; the flavors improve with 15-20 minutes of resting.

For charcoal: bank hot coals to one side. For gas: preheat all burners to high, then turn one side to medium-low. Clean and oil the grates thoroughly - this is crucial for fish. The grill should be hot enough that you can only hold your hand 5 inches above for 2-3 seconds.

Brush the remaining 2 tablespoons olive oil on the fish. Lay the snapper diagonally across the hot grates and do not move for 5 minutes. The skin should release naturally when properly seared. Flip carefully with a fish spatula and grill the second side until the flesh flakes at the thickest part and registers 135-140F, about 4-5 minutes more.

Transfer the fish to a platter and let rest 3 minutes. The internal temperature will climb to 145F. Spoon half the salsa verde over the fish and serve the rest on the side. Encourage guests to pull tender pieces of meat from the head and cheeks - the best bites.