Boiling kale before sautéing it is key to making it tender. If the stems and center veins are very tough, Marcia Kiesel advises removing them: Fold the kale leaves in half with the vein side out, then pull up on the stems. Other creative ideas for braised greens are Kale and Water Chestnuts in Curried Cream and Kale with Currants, Lemon and Olives. More Recipes for Greens
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.
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.
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.
The kale can be covered and refrigerated overnight; reheat gently over moderate heat, adding a few tablespoons of chicken broth as necessary.