Bring a large pot of heavily salted water to a boil. (A few five-finger pinches of salt should do!) Cook pasta, stirring occasionally, 1 minute shy of package directions.
In a small mixing bowl, whisk together the eggs, 2/3 cup of Parmesan, a pinch of salt, and lots of pepper until no streaks remain.
Remove the sausage from the casings, and cook in a pan over medium-high heat, breaking up the sausage into small crumbles with a wooden spoon. Cook until the sausage is no longer pink and starting to get browned and crispy on the edges. Drain off all but 1 Tablespoon of fat.
Add the greens to the sausage mixture, and sauté just until the greens are wilted.
When the pasta is finished cooking, reserve about 1 cup of pasta water, and drain. Return the bucatini to the pot off the heat, and add the egg-Parm mixture with a splash of pasta water, stirring the pasta quickly and constantly to prevent the eggs from scrambling. Add more pasta water as needed until the sauce is the consistency of heavy cream.
Add the sausage-greens mixture to the pasta with a few turns of pepper. Taste, and add more salt and pepper, if needed.
Serve the bucatini carbonara with grated or shredded Paremsan. Enjoy!