Cut the stems off the chard and coarsely chop the leaves. Set the leaves aside and discard the stems or save them for another use.
Put the butter and sage in a large, deep sauté pan over medium heat and cook until the sage lightly browns, the butter turns golden, and the milk solids from the butter lightly brown on the bottom of the pan, 5 to 7 minutes.
Bring a large pot of salted water to a boil. Add the potatoes and cook until barely tender (a fork inserted should meet some resistance), 3 to 4 minutes. Using a slotted spoon, remove the potatoes to the pan with the browned butter; reserve the pot of boiling water for the pasta. Add the shallot to the pan and cook over medium heat until the potatoes are tender, 3 to 4 minutes longer.
Add the pasta to the boiling water and cook until tender yet firm, 4 to 5 minutes.
Meanwhile, add the oil, chard, and 1 cup of the pasta cooking water to the pan of potatoes. Cook until the chard wilts and the liquid in the pan reduces slightly in volume, 3 to 4 minutes.
Drain the pasta and add it to the pan along with the Fontina and Gruyère, stirring until the cheese forms a creamy sauce. Season with the salt and pepper, and serve on a large platter or divide among warm pasta dishes.
Tip: If you don’t have Gruyère, use good-quality Parmesan instead.