Bring a large pot of water to a boil.
In a large skillet, heat the EVOO, two turns of the pan, over medium. Melt the butter into the oil. Add the shallots, garlic and thyme; season with salt and pepper. Cook, stirring frequently, until light-caramel color, 15 to 20 minutes.
Salt the boiling water and cook the pasta to al dente; add the kale for the last 4 minutes of cooking time. Ladle out about a cup of the starchy cooking water and add to the shallot sauce. Drain the pasta and return to the pot. Add the sauce, tossing with tongs 1 or 2 minutes for the flavors to absorb. Adjust the salt and pepper and toss in the cheese and parsley.