In a large saucepan, bring vegetable broth to boiling. Meanwhile, in a small bowl, stir together milk and polenta. Add polenta mixture in a slow, steady stream to vegetable broth, stirring constantly; reduce heat. Cook and stir until bubbly; cook and stir for 3 to 5 minutes more until polenta is thickened. Stir in cheese and diced green chilies until well mixed; set aside.

In an extra-large skillet, cook sausage, mushrooms and onion over medium-high heat until meat browns and onion is tender. Drain and discard liquid. Stir in polenta mixture until well mixed.

In a coated 2 quart au gratin dish, spread polenta-meat mixture. Cover baking dish with a sheet of greased or nonstick foil; store overnight in the refrigerator.

While oven preheats to 375° , use the bottom of a 1/3-cup measuring cup coated with olive oil to make four indentations in the polenta mixture. Bake, covered, for 30 minutes. Reduce oven temperature to 325°. Remove baking dish from oven; remove foil. Break one egg into a small dish. Carefully slide the egg into an indentation. Repeat with remaining eggs. Return to oven.

Bake, uncovered, in the 325° oven for 20 minutes more or until the eggs are almost set. Let stand on wire rack 5 to 10 minutes (eggs will continue to cook). If desired, season with salt and pepper, snipped cilantro and serve with hot pepper sauce.