Bring a large pot of water to a boil. Fill a bowl with ice water. When the water comes to a boil, salt generously and add the greens (you may have to do this in two batches). After the water returns to a boil, boil two to four minutes until the greens are tender. Using a deep-fry skimmer or slotted spoon, transfer them to the ice water. Do not drain the hot water in the pot, as you’ll use it to cook the pasta. Drain the greens, squeeze dry and chop.
Heat the oil over medium heat in a large, heavy nonstick skillet. Add the garlic, cook for about a minute just until fragrant, and stir in the greens. Toss in the hot pan for about a minute, just until the greens are lightly coated with oil and fragrant with garlic. Season with salt and pepper, and remove from the heat.
Place the ricotta in a large pasta bowl. Bring the water for the pasta back to a boil, and add the pappardelle. Cook al dente. Ladle 1/2 cup of the cooking water from the pasta into the ricotta and stir together. Drain the pasta, and toss with the ricotta, greens and cheeses. Serve at once.
Tip: Advance preparation: The greens, prepared through step 2, will keep for up to three days in the refrigerator.