Preheat the oven to 400 degrees F. and place a rack in the upper middle position.
In a 9x13-inch baking dish, place the potatoes, carrots and onion. Toss with 1 tablespoon of oil and salt and pepper to taste.
In a small bowl, mix the butter, lemon zest, fresh herbs and garlic together with a fork. Set aside.
Pat the game hens dry with a paper towel, then rub the herb butter over the game hens coating on all sides. Salt and pepper liberally and lay all four pieces on top of the veggies.
Roast for 25 minutes. Remove from the oven and baste. Then continue to roast for another 20 minutes, until the juices run clear when pricked with a fork. Allow the hens to rest for at least 5 minutes before serving.