How to Make It
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.