In a large pot, cook pasta in salted boiling water according to package directions; drain, reserving ¾ cup pasta cooking water. Return pasta to pot; cover to keep warm.
Meanwhile, in a very large skillet heat olive oil over medium-high heat. Add Swiss chard stems, garlic and salt. Cook and stir for 2 minutes.
Add the tomato paste. Cook and stir for 1 minute more.
Gradually whisk in ½ cup of the pasta cooking water until smooth. Bring to a simmer.
Add cooked pasta, chard leaves, basil and vinegar to sauce; stir to combine.
If desired, stir in up to ¼ cup additional cooking liquid to reach desired consistency.
Season with black pepper. Divide among serving dishes. Sprinkle with cheese and additional pepper, if desired.
