Bring a large pot of water to a boil. Season with salt.
In a large saute pan, 2 tablespoons olive oil over medium heat along with the sausage and 'nduja and cook, breaking apart the meat into smaller pieces, until the sausage is browned, about 10 minutes. Add the garlic and mushrooms and increase the heat to medium-high.
Cook until the mushrooms begin to brown, then give them a stir and season with salt and pepper. Add the wine and simmer for 1 minute. Add the kale and continue to cook until slightly softened, 2 to 3 minutes longer. Add the pasta to the water and cook to al dente.
Strain the pasta, reserving a small amount of the water in a bowl. Add the pasta to the saute pan and bring to a simmer. Add pasta water by 1/4 cup as needed if the pasta is dry. Toss, cooking until the pasta begins to get coated and the sauce begins to thicken. Add the remaining tablespoon olive oil. Add the butter, then remove the pan from the heat and add half the cheese. Stir the mixture until thickened and glossy. If too thick, add a splash of the pasta water. Season to taste.
Divide between bowls and garnish with the remaining cheese.