In a large pot, heat the extra-virgin olive oil over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes. Toss in the minced garlic and sauté for another minute until fragrant.
Stir in the tomato paste and crushed red pepper flakes, cooking for an additional 2 minutes to deepen the flavors.
Pour in the San Marzano diced tomatoes along with their juices and the chicken broth. Stir well to combine. Season with salt and pepper to taste.
Increase the heat and bring the mixture to a gentle simmer. Once simmering, reduce the heat to low and let it cook for about 10 minutes.
Add the penne pasta and cook according to package instructions, around 8-10 minutes, until al dente.
Stir in the shredded rotisserie chicken, mozzarella cheese, and grated Parmesan cheese. Mix until the cheese is melted and the soup is creamy.
Remove from heat and stir in the chopped fresh basil. Serve hot, garnished with additional cheese and basil if desired.