Cut white stems of chard into half-inch pieces (for cooking). Begin to boil these first, as they take longer to cook. After 2 minutes, add the rest of the greens. Boil for another 2 minutes, then quickly drain and soak them in ice water. Once ice dissolves, drain well.
Combine scallions, 1 clove of garlic and 1 tablespoon of olive oil in a saucepan. Sauté on medium-high heat for 2 minutes, until golden, then add tomatoes and simmer for 20 minutes, uncovered.
Bring a large pot of salted water to a boil. Cook linguine to al dente. Drain.
In a separate saucepan, add 2 tablespoons of olive oil, remaining chopped garlic and boiled Swiss chard. Season with salt and pepper. Sauté together for 3 to 4 minutes, then add tomatoes, basil and the fresh cream. Reserve a few basil leaves for garnishing. Add hot pasta to saucepan and sauté together for about 2 minutes. Turn off the heat, add butter, half of the Parmigiano and ground pepper.
Serve topped with Parmigiano-Reggiano and chopped basil leaves.