Bring a large pot of salted water to a boil. Add 1/2 pound whole wheat spaghetti and cook according to package directions until al dente. Before draining, reserve 1 cup of the starchy pasta water, then drain the pasta.
While the pasta cooks, heat a large skillet over medium heat. Add 2 tablespoons unsalted butter and let it melt.
Stir in 1 teaspoon freshly cracked black pepper and toast in the melted butter for about 60 seconds, stirring so the pepper becomes fragrant.
Add the chopped kale (stems removed) to the skillet and stir to coat with the butter and pepper. Cook until the kale begins to wilt and is bright green, about 2–3 minutes.
Pour in about 3/4 cup of the reserved pasta water and bring the mixture to a gentle simmer.
Add the remaining 2 tablespoons unsalted butter and the drained pasta to the skillet. Toss the pasta, kale, butter, and water together until combined and the kale continues to soften.
Reduce the heat to low. Stir in 3/4 cup finely grated parmesan cheese and 1/4 cup finely grated pecorino cheese, tossing continuously until the cheeses melt and form a creamy sauce that coats the pasta. Add more of the reserved pasta water as needed to loosen the sauce.
Serve immediately topped with parmesan shavings and additional freshly cracked black pepper.