1. Preheat the oven to 350°. In a heavy 10- to 12-inch skillet, heat the oil over moderate heat until a light haze forms above it. Drop in the onions and garlic and, stirring frequently, cook for about 5 minutes, until they are soft and transparent but not brown.
2. Stir in the beef and, mashing it with a spoon to break up any lumps, cook until all traces of pink disappear. Add the tomatoes, chilies, salt and a few grindings of pepper.
3. 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.
4. 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. Set the pan in the middle of the oven, and pour in enough boiling water to come about 1 inch up the sides of the papayas.
5. Bake for 1 hour. Then sprinkle each shell with 1tablespoon of the cheese, and bake for 30 minutes, or until the papaya shows no resistance when pierced with the point of a small knife and the top is delicately browned.
6. To serve, transfer the papaya shells to a large heated platter and sprinkle them with the remaining 2 tablespoons of cheese.
