Bring a large pot of water to a boil over medium-high heat.

Heat olive oil in a large skillet over medium heat until shimmering. Add anchovy fillets and their oil. Cook until the anchovies dissolve into a loose paste, about 2 minutes.

Add sliced cabbage and kosher salt. Stir well, cover, and cook for 5 minutes. Stir and cook for another 5 minutes. Uncover and continue to cook until the cabbage is very soft, about 5 more minutes.

While cabbage is cooking, add spaghetti and remaining kosher salt to the boiling water. Cook according to package directions until just al dente, about 8 to 12 minutes.

In a medium bowl, whisk together eggs, egg yolks, Pecorino Romano cheese, black pepper, and a pinch of kosher salt. Slowly stream in pasta cooking water while whisking constantly.

Transfer spaghetti to the skillet, add some pasta water, lemon zest, and lemon juice. Toss until well-coated.

While tossing, slowly pour egg mixture into the skillet and cook on the lowest heat until the sauce thickens and coats the pasta, about 1 to 2 minutes.

Taste and season with more salt and pepper as needed. Serve immediately with chopped parsley, more grated Pecorino, and black pepper.