Preheat the oven to 475 degrees.
Pulse the tomatoes and their juices in the food processor until
coarsely ground, about 12 pulses.
Cook the oil, garlic, red pepper flakes, and 1/4 teaspoon of salt in a
12-inch skillet over medium-high heat until fragrant, about 1
minute. Stir in the tomatoes, water, 1 cup of the broth, and the
pasta. Bring to a vigorous simmer, cover, and cook until pasta is
tender, 15 to 18 minutes, stirring often.
Stir in 3/4 cup of the Parmesan, 1/4 teaspoon of pepper, and the
chopped chicken. Taste and adjust the seasonings if needed. If the
pasta seems dry, stir in the remaining 1/2 cup of broth. Transfer to
a baking dish and sprinkle with mozzarella and remaining
Parmesan. Bake 5 to 10 minutes, or until the cheese is melted and
bubbly. Sprinkle with basil and serve.