Pulse the tomatoes and their juices in the food processor until coarsely chopped, 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, oregano 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 buttered oven-proof 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.