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Bring a large pot of salted water to a boil. Add the dry pasta and cook according to package instructions until al dente. Reserve 1/2 cup of pasta water, then drain the pasta and set aside.
While the pasta cooks, heat the olive oil and half of the butter (20 g) in a large skillet over medium heat. Add the minced garlic and sauté for about 1 minute until fragrant.
Add the thinly sliced cabbage to the skillet. Season with 1/2 tsp salt and 1/4 tsp black pepper. Cook, stirring occasionally, for 6-8 minutes until the cabbage is soft and slightly caramelized.
Push the cabbage to one side of the skillet. Crack the eggs into the empty side and scramble them gently until just set, then mix them into the cabbage.
Add the drained pasta to the skillet along with the remaining butter (20 g). Toss everything together, adding a splash of reserved pasta water if needed to create a silky sauce.
Season with the remaining salt and black pepper to taste. Serve hot, garnished with extra black pepper if desired.