Place 2 tablespoons of the butter in a large, heavy-bottomed pan over medium heat. When melted, add the shallot and garlic. Cook over low heat without letting the garlic and shallots color, about 3 to 4 minutes. Add the kale and continue to cook for a few more minutes until wilted. If it won’t all fit in the pan, just add what’s left after it’s cooked down a bit.
Meanwhile, in a small pan set over medium-high heat, melt 1 tablespoon of butter. Add the mushrooms and cook until softened and cooked through, about 5 minutes. If the mushrooms become too dry, add more butter. Stir in the soy sauce, cook another minute and turn off the heat.
Once the kale is wilted and soft, increase the heat to medium-high, add the vermouth and cook until it’s just evaporated, about 1 minute. Add the cream and miso, stirring until completely incorporated. Reduce heat to medium and cook about 2 more minutes until the sauce reduces slightly and tightens up the around the kale. Taste for seasoning, add salt and pepper if you like (but don’t forget the mushrooms have soy sauce). Place the kale on a warm platter and scatter the mushrooms over the top. Serve immediately.