Preheat the oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish.
Rinse the quinoa in a fine mesh sieve and let soak briefly in warm water, then drain.
Bring 2 cups chicken broth to a boil in a small saucepan, add the drained quinoa, cover, reduce to a simmer, and cook until the liquid is absorbed, about 12–15 minutes.
Remove the quinoa from heat and let it sit, covered, for 5 minutes; fluff with a fork and set aside.
In a medium saucepan over medium heat, combine crushed tomatoes, tomato paste, and balsamic vinegar; bring to a gentle simmer.
Stir in the half and half, crushed red pepper (if using), and salt and pepper to taste; heat through and remove from the heat.
In a large bowl, combine the diced cooked chicken, cooked quinoa, grated Parmesan, and 1/2 cup of the shredded mozzarella.
Pour the warm tomato sauce into the chicken-quinoa mixture and stir until evenly combined.
Transfer the mixture to the prepared baking dish, top with the remaining mozzarella and the halved grape tomatoes.
Bake for 12–15 minutes, until the cheese is melted and the casserole is heated through.
Remove from the oven, sprinkle with chopped fresh basil, then serve.