Stirring occasionally, cook briskly until most of the liquid in the pan has evaporated and the mixture is thick enough to hold its shape almost solidly in the spoon.
Taste for seasoning.
Spoon the meat mixture into the papaya shells, spreading and smoothing the filling with a spatula.
Place the shells side by side in a shallow roasting pan.
Pour enough boiling water in the pan to come about 1 inch up the sides of the papaya.
Bake for 1 hour then sprinkle each shell with 1 tbsp of the cheese and bake for 30 minutes, or until the top is delicately browned.
Top shells with remaining cheese when ready to serve.