Rub the insides of Cornish hens with salt and pepper and 3 minced garlic cloves.

In a bowl, combine 2 tablespoons olive oil, paprika, chili powder, cumin, remaining minced garlic, 1/4 cup chicken stock and stir well to combine. Use 2/3 of this mixture to rub all over the outside of Cornish hens. Sprinkle the desired amount of salt over the outside of the hens. Sprinkle a very small amount of additional paprika over the hens (for color).

Toss carrots, onions, and sweet potatoes in the remaining 1/3 of the mixture to coat. Sprinkle with salt and pepper.

Transfer vegetables to the large roasting pan. Place hens breast sides up on top of vegetables. Add a little bit of chicken stock on the bottom of the pan to prevent vegetables from burning.

Transfer hens and vegetables to a platter. Make sure to pour juices from the hen cavities into a roasting pan. Cover hens with foil. Transfer the sauce from the roasting pan into a smaller saucepan and boil it down a little bit to thicken, if desired. You can also dissolve a little bit of cornstarch in water and add it to the boiling sauce to thicken it. After hens sit under the foil, pour juices from rested hens back into the sauce you just made. If desired, add more chicken stock to the thickened sauce.

Serve Cornish hens whole or cut in half.