In a large saute pan over low heat, melt butter. Add flour and cook for 2 to 3 minutes, taking care not to brown the flour. Slowly add hot milk, using a wire whisk to stir. Cook the roux into the milk over low simmer for 3 to 4 minutes, until mixture has thickened. Add the salt, pepper and cayenne. Stir in the cheese and the green onions. Remove from the heat when the cheese has fully melted.
Using a large baking dish or individual ramekins, pour the hot sauce into the dish(es). Break the eggs, one at a time, into a small, shallow fruit dish and slide them, one at a time, into the sauce. Place in the oven and bake for 7 to 8 minutes or until the whites and yolks have begun to set. Serve immediately.