Bring a 6-quart pasta pot of water to a boil and salt it generously with the 2 tbsp kosher salt.
Add the fettuccine and cook according to package time until al dente. Scoop out 1 cup / 240 ml of pasta water before draining, then drain the pasta.
While the pasta cooks, melt the butter in a 12-inch skillet over medium heat.
Once the butter foams, add the whole sage leaves. Swirl the pan and cook for 2 to 3 minutes, until the foam subsides, the butter turns deep amber, and the leaves crisp at the edges. Watch closely, browned butter can turn from nutty to burnt in under 30 seconds.
Pull the skillet off the heat as soon as the butter smells nutty. Add a splash of the reserved pasta water to stop it from browning further.
Add the drained fettuccine directly to the skillet and toss with tongs, adding more pasta water a splash at a time until the sauce coats the noodles and looks glossy.
Add half the finely grated Parmigiano Reggiano and the black pepper, and toss again until the cheese melts into the sauce.
Plate the pasta and top with the remaining finely grated Parmigiano Reggiano and any extra crisp sage leaves. Serve right away.