Using kitchen shears or a sharp chef knife, cut out backbone of each hen, and split through each breastbone. Trim any excess fat and skin.
Brush oil onto large shallow pan (15 x 10 x 1-inch) and place in oven. Heat oven to 400°F.
In small bowl, whisk together lemon juice, honey, thyme and oregano. Season game hen halves with salt and pepper. Once pan has heated, carefully remove from oven. Place hens, skin side down, into pan. Brush with half of the honey mixture.
Bake at 400°F for 30 minutes. Remove pan from oven and turn over hens. Brush with remaining honey mixture. Return to oven and bake an additional 20 to 30 minutes or until internal temperature of meat registers 180°F on an instant-read thermometer. Transfer to platter.
Drain pan drippings into a fat separator, or skim off as much fat as possible. Pour into a gravy boat, and serve alongside hens.