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

Heat olive oil in a large skillet over medium heat; add anchovy fillets and cook until dissolved.

Add cabbage and kosher salt, cover and cook undisturbed for 5 minutes.

Stir, cover, and cook for another 5 minutes, then uncover and continue to cook until soft.

Cook spaghetti according to package directions, adding remaining salt.

In a bowl, whisk eggs and yolks, then combine with cheese, black pepper, and a pinch of salt.

Slowly add pasta cooking water to egg mixture while whisking.

Transfer spaghetti to skillet, tossing with cabbage mixture, lemon zest, and lemon juice.

Slowly pour egg mixture into skillet while tossing pasta, and cook on low heat until thickened.

Adjust seasoning with salt and pepper; serve with parsley and additional cheese if desired.