Bring water, salt and oil to a boil in a large saucepan.
In the meantime, cut the chard leaves into noodle-thick strips. Dice the Gorgonzola and mozzarella very finely, grate the parmesan. Chop the onion and garlic very finely, sauté in half the butter. Pour in the cream and broth. In an open pot, reduce by about a third in about ten minutes. Add half of the Gorgonzola and Parmesan to the sauce and let it melt, seasoning with salt if necessary. Cook the pasta, cook the chard strips for the last two minutes. Then drain everything through a sieve. Drain the chard well and toss in the remaining butter. Mix the pasta with the chard and serve with the remaining cheese and on preheated deep plates. Pour the sauce over them.