Pat the chicken dry on a cutting board and season with 1/2 teaspoon salt, 1/4 teaspoon paprika, and 1/4 teaspoon black pepper.
Heat 2 tablespoons olive oil in a Dutch oven or large stockpot over medium-high heat. Add the chicken and cook until the outside is opaque but not fully cooked through, about 2-3 minutes; transfer the chicken to a plate.
Add 1 tablespoon olive oil to the pot if needed, then add the chopped onion, celery, and carrots. Cook over medium-high heat until softened, about 5–7 minutes.
Stir in the minced garlic and red pepper flakes and cook for 1 minute until fragrant.
Pour in 3 cups water, crushed tomatoes, fire-roasted tomatoes (undrained), and tomato paste. Add the dried basil, dried parsley, chicken bouillon, sugar, oregano, thyme, 1/4 teaspoon salt, and 1/4 teaspoon black pepper. Stir to combine and bring to a boil.
Reduce to a strong simmer, add the partially cooked chicken back to the pot, then add the dry pasta and stir to combine.
Cover and cook, stirring occasionally and briefly lifting the lid to stir, until the pasta is al dente, about 10–15 minutes. Check frequently near the end and add a splash of water if the sauce is evaporating too quickly.
Reduce heat to low. Stir in 1/2 cup Parmesan and 1/4 cup of the mozzarella until melted, then stir in the heavy cream. Sprinkle the remaining 3/4 cup mozzarella evenly over the pasta.
Cover and cook 2–3 minutes more until the top mozzarella is melted. Serve immediately.
