Coat each Cornish hen with 2 tablespoons of the butter and sprinkle with salt and pepper. Into the cavity of each bird, insert a lemon half, an onion quarter, half a garlic head and 3 thyme sprigs. Set aside.
Spread the carrots and potatoes in an even layer in a 9-by-13-inch baking dish or roasting pan. Add the chicken broth and then the prepared Cornish hens. Roast until the juices run clear and the hens are golden brown, 45 minutes to 1 hour.