Heat a large pot of salted water to boiling.
Heat the olive oil in a very large sauté pan or skillet over high heat. Add the onions and cook until tender. Stir in half the sage. If you don’t mind dirtying another pan, you may brown the potatoes separately. Add the potatoes (browned or not), Swiss chard, green beans, and garlic; cook until the greens are wilted, about 3 minutes.
Meanwhile, add the pasta to the boiling water and cook until tender, about 2 minutes if the pasta is thin.
Add the remaining sage, the cheeses, and salt and pepper to taste to the greens mixture. Drain the pasta and transfer to a pasta bowl. Add the greens mixture and quickly toss to combine.