Heat oven to 400° F. Season the lamb with 1 teaspoon salt and ½ teaspoon pepper and place in a roasting pan.
In a small bowl, combine the garlic, tarragon, and 1 tablespoon of the oil. Rub over the lamb.
Roast the lamb, 25 to 30 minutes for medium-rare (internal temperature 125° F). Transfer to a cutting board. Let rest for 5 minutes before slicing.
Meanwhile, heat the remaining 3 tablespoons oil in a skillet over medium-high heat. Add the onion and cook until soft, about 4 minutes.
Add the beans, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring, until heated through, 3 to 4 minutes. Serve with the lamb.