Whisk together lemon juice, salt and pepper, then slowly add olive oil in a stream, whisking until combined. Add oregano and set aside.
Along the middle of each fish make 1-inch long diagonal slashes at 2-inch intervals on both sides of the fish. Brush fish all over with vegetable oil and season with 1 teaspoon each of salt and pepper (total).
Season the fish cavity with salt and pepper. Insert 2 lemon slices and 2 sprigs of oregano in each fish. Using kitchen string tie fish closed at 2-inch intervals.
Oil grill rack then place branzino on the grill. Close lid and cook for 6 minutes. Turn fish over using 2 metal spatulas. Cook for another 6 minutes or until cooked through.
Transfer fish to platter, remove string and garnish with more lemon. I say let each guest cut off the head and remove the bones of their own fish.
Serve fish with dressing.