Cook pasta according to package instructions. Drain, reserving 1/2 cup of pasta water for finishing sauce.
Meanwhile in a large saucepan, heat olive oil over high heat. Add sweet potatoes and cook until they begin to brown, about 5 minutes. Reduce heat to medium and stir in onions and garlic. Cook for another 2 minutes, or until onions begin to brown. Add mushrooms and cook until tender; season with salt and pepper.
Bring another saucepan of salted water to a boil. Add kale and cook for 2-3 minutes until brightly colored; drain. Toss hot pasta and blanched kale leaves with sautéed vegetables in saucepan. Add pasta water, if desired, for sauce consistency. Cook together for 2 minutes on medium heat. Serve with grated Parmesan.