Place the game hen and ginger into a 4-quart pot and cover with 5 cups chicken broth, reserving the remaining cup to thin out porridge, if desired. Bring to a boil, and then reduce to a simmer. Skim any foam or impurities from the top. Add the rice, cover, and cook on low for 1 hour; the chicken will become tender and the rice will expand and form a porridge-like consistency. Remove the game hen, cool briefly, and shred the meat. Discard skin and bones and add the shredded meat back to the pot. Remove the ginger. If a thinner soup is desired, add the additional chicken broth to desired consistency. Season with fish sauce and freshly ground black pepper. Serve hot, sprinkled with green onions and cilantro.