Fill a large pot with water and bring to a boil, making sure to salt the water enough that you can taste it.
Meanwhile, 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.
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, 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. Add the red pepper flakes and season with salt and pepper. Keep warm.
Add the pasta and cook until soft, not al dente. Reserve 1/2 cup pasta water. Combine the yogurt with 1/4 cup cooking water and mix well. Add more of the reserved pasta water as needed to get the sauce to your thickness. Drain the pasta and toss with the yogurt mixture.
Serve the pasta immediately, with a generous amount of cheese and topped with the creamed spicy greens.