Place the game hen breast side down on a cutting board. Using scissors or poultry 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.
Spray the bottom of the panini press with the nonstick cooking spray. Place the bird, skin side up onto the press, pray the skin side with non-stick cooking spray and close the lid. Top the press with a 10-pound weight and cook for 8 to 10 minutes or until the skin is golden brown.