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4 pounds kale, tough stems discarded
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3 tablespoons unsalted butter
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3 tablespoons extra-virgin olive oil
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1 medium onion, cut into 1/4-inch dice
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1 pound shiitake mushrooms, stems discarded and caps sliced 1/3 inch thick
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Salt and freshly ground pepper
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1/2 cup low-sodium chicken broth
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3 tablespoons soy sauce
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2 teaspoons toasted sesame oil
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In a large pot of boiling water, cook the kale until tender, about 5 minutes. Drain and when the kale is cool enough to handle, gently squeeze out the excess water. Coarsely chop the kale.
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Meanwhile, in a large, deep skillet, melt the butter in the olive oil. Add the onion and cook over moderate heat until softened and golden brown, about 7 minutes. Add the shiitake, season with salt and pepper and cook over moderately high heat, stirring, until softened, about 3 minutes. Cover and cook, stirring occasionally, until the shiitake are richly browned, about 5 minutes longer.
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Add the chicken broth and kale to the skillet and toss. Cover and cook over moderate heat, stirring a few times, until the kale is sizzling, about 3 minutes. Stir in the soy sauce and sesame oil, season with salt and pepper and serve.
Make Ahead
The kale can be covered and refrigerated overnight; reheat gently over moderate heat, adding a few tablespoons of chicken broth as necessary.