Heat oven to 500 degrees. Bring a large pot of salted water to a boil. Add pasta, and cook for 4 minutes. Drain in a colander.
While pasta is cooking, place cream, tomatoes, cheeses, salt, and basil in a large bowl. Stir to combine. Add cooked pasta to cheese mixture, and toss to combine.
Divide among six to eight shallow ceramic gratin dishes (1 1/2 to 2 cups in capacity). Dot with butter, and bake until bubbly and brown on top, 7 to 10 minutes. Sprinkle with pecorino Romano and scallions. Serve immediately.