Bring a large pot of well-salted water to a rolling boil and cook the bucatini until al dente according to package directions.
While the pasta cooks, whisk the eggs in a mixing bowl, then stir in the chopped parsley, chopped basil and the grated Parmesan. Add a pinch of salt and black pepper and whisk to combine.
When the pasta is done, reserve a small cup of pasta cooking water, then drain the pasta and return the hot pasta to the pot off the heat.
Working quickly, pour the egg-herb-Parmesan mixture over the hot pasta and immediately toss vigorously with tongs or a wooden spoon so the eggs cook from the residual heat and form a creamy sauce; toss for 2–3 minutes.
Add the butter and toss once more until melted and fully incorporated; if needed, add a splash of reserved pasta water to loosen the sauce.
Taste and adjust seasoning with salt and freshly ground black pepper, then serve immediately.