Using a mini food processor, chop together the garlic, parsley, tarragon, chives, lemon peel and 1 teaspoon salt. With the machine on, drizzle in 3 tablespoons olive oil and process until a coarse paste forms.
Preheat the oven to 450 degrees . Gently slide your fingers between the skin and flesh of the chicken breasts and legs. Place breast side down on a cutting board, with the large cavity facing you. Using kitchen shears, start at the open cavity and cut down each side of the backbone; discard the backbone. Open the chicken like a book, then flip over. Press down firmly on the top fleshy part of the breast until you hear the bone crack.
Using your fingers, stuff the herb mixture under the skin, spreading it evenly. Rub the remaining 1 tablespoon olive oil on the skin and underside of the chicken; season with salt and pepper. Place breast side up in a roasting pan and cook until the skin is golden and crispy or until an instant-read thermometer registers 165 degrees , about 45 minutes. During the last 10 minutes, cover loosely with foil. Let the chicken rest for 15 minutes before carving.