Heat a panini press on high.
Place the game hen, breast-side down, on a cutting board. Using kitchen shears, cut from the neck to the tailbone to remove the backbone. Once you remove the backbone you will be able to see the inside of the bird. Make a small slit in the cartilage at the base of the breastbone to reveal the keel bone. Grab the bird with both hands on the ribs and open up like a book, facing down towards the cutting board. Remove the keel bone. Cut small slits in the skin of the bird behind the legs and tuck the drumsticks into them in order to hold them in place. Season on both sides with salt and pepper.
Coat the bottom of the panini press with nonstick cooking spray. Place the bird, skin-side up, onto the press, coat the skin side with nonstick cooking spray and close the lid. Top the press with a 10-pound weight and cook until the skin is golden brown, 8 to 10 minutes.
Allow to rest 3 minutes before serving.