Bring a large pot of 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 large heatproof bowl until smooth, then stir in the chopped parsley, chopped basil, grated Parmesan, and a pinch of salt and pepper.
Reserve about 1/2 cup of the hot pasta cooking water, then drain the pasta in a colander.
Return the hot pasta to the empty pot off the heat, or place it in the heatproof bowl; working quickly, add the egg and herb mixture and begin tossing or stirring vigorously so the residual heat thickens the eggs into a creamy sauce. Add a little reserved pasta water as needed to loosen the sauce.
Add the butter and toss once more until melted and evenly coating the pasta; continue tossing for 2–3 minutes until the sauce is smooth and the eggs are cooked through.
Taste and adjust with salt and freshly ground black pepper, then serve immediately.