In a heavy 4-quart saucepan, place chanterelles, porcini, oyster, and yellowfoot mushrooms, red onion, chiles, scallions, garlic, white wine, and tomato sauce and bring to a boil. Lower heat and simmer, partially covered, for 20 minutes; mixture should be thick like a ragu. Season with salt and pepper and pour into a large shallow serving bowl.
Toss puntarelle in a bowl with the anchovies, lemon zest and juice, and oil until well coated. Arrange puntarelle mixture over mushrooms and serve with grilled bread on the side.