In a medium saucepan of boiling water, cook the onions over moderate heat until just tender, about 8 minutes. Drain and peel.
In a large skillet, melt the butter in 1/4 cup of the olive oil. Add the onions and cook over moderate heat until browned, about 4 minutes per side. Add the garlic and cook, stirring, until fragrant, about 3 minutes. Season with salt and pepper.
In each of 2 large skillets, heat 2 tablespoons of the olive oil until shimmering. Add the chanterelles to 1 skillet and the shiitakes to the other, season with salt and pepper and cook over moderate heat until the mushrooms release their liquid, about 4 minutes. Continue cooking until the chanterelles are tender and the shiitakes are browned, about 5 minutes longer. Transfer to a bowl.
In 1 of the skillets, cook the oyster mushrooms in the remaining 2 tablespoons of olive oil. Add the stock and cream and bring to a boil. Stir in the onions, chanterelles and shiitakes and simmer over moderately low heat, stirring gently, until heated through, about 4 minutes. Season with salt and pepper and serve.
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