For the chicken: Preheat the oven to 425 degrees F.
Finely grate the zest of one of the lemons and cut that lemon in half. Rub the chicken with salt and pepper inside and out, then rub the lemon zest all over the outside. Stuff the cavity with the halved lemon, tarragon and garlic. Put the chicken in a baking dish or small roasting pan. Cut the remaining lemon in quarters and add it to the dish along with 3/4 cup water. Brush the outside of the chicken with the butter.
Roast until golden brown and an instant-read thermometer inserted into the center of the thigh registers 165 degrees F, about 1 hour. Transfer the chicken to a cutting board and pour the pan juices into a small bowl. Let the chicken rest 10 minutes. Use a small ladle to remove the fat that rises to the top of the pan juices.
For the chimichurri: Meanwhile, finely chop the parsley, tarragon, onions and garlic and add to a bowl. Stir in the olive oil, vinegar, red pepper flakes and 1/4 teaspoon salt.
Carve the chicken and serve with the pan juices and chimichurri.