Heat 4 tablespoons oil in a large cast iron skillet over medium high heat. Add mushrooms, thyme sprigs (if using), and garlic cloves and cook undisturbed until mushrooms are golden brown, 4 minutes. Mix around and cook until mushrooms have shrunk and all sides are golden brown, another 3 minutes. Season with salt and pepper, to taste. Remove with a slotted spoon and transfer to a paper towel.
In the same skillet, heat 2 more tablespoon oil. Add white beans (they might splatter at first, so add with caution) and rosemary sprigs (if using), and cook until golden brown, 4 minutes. It's ok if their skins rub off the bean--as long as everything is nice and crispy. Season with salt and pepper, to taste. Remove with a slotted spoon and transfer to a paper towel.
Meanwhile, bring a large pot of heavily salted water to a boil and cook noodle to al dente. Reserve 1 1/2 cups pasta water and drain.
In the same cast iron, reduce heat to medium low and heat remaining 1 tablespoon oil and cook shallot until softened, 1 minute. Add cream, cup pasta water, lemon zest and juice. Add pasta and toss to coat. Add butter and kale. Cook until kale is wilted, 5 minutes (you may need to add more pasta water--the sauce should look smooth and glossy). Add Parmesan and season to taste with salt and pepper. Add mushrooms and beans and cook until warmed through, 3 minutes. Top with more Parm and serve immediately.