Bring a large pot of generously salted water to a boil and cook the pasta until just before al dente. Before draining, set aside 1 cup of the pasta cooking water. In a large bowl, toss the cubed chicken with 2 tablespoons of oil from the jar of roasted tomatoes (or 1 to 2 tablespoons extra-virgin olive oil), Italian seasoning, paprika, ¾ teaspoon salt, and ½ teaspoon black pepper. Heat a large oven-safe skillet over medium heat, add the chicken, and sauté until fully cooked, about 8 minutes. Remove the chicken and transfer it to a bowl. In the same skillet, add the shallots along with an extra drizzle of oil if needed, and cook until softened, about 3 minutes. Stir in the garlic and crushed red pepper flakes, cooking until fragrant, about 1 minute. Add the roasted tomatoes (or sun-dried tomatoes) and spinach, cooking until the spinach wilts, about 2 to 3 minutes. Season with the remaining ½ teaspoon salt. Mix in the pesto and heavy cream, then add the pasta along with ½ cup of the reserved pasta water. Stir continuously until the sauce thickens and coats the pasta, adding more pasta water as necessary until the sauce is creamy and thick—usually the full 1 cup is needed. Stir in 1/3 cup of the Parmigiano-Reggiano cheese and the cooked chicken. Preheat your oven broiler to HIGH, then sprinkle the torn mozzarella and the remaining Parmigiano-Reggiano cheese evenly over the pasta. Place the skillet under the broiler and broil until the mozzarella is melted and bubbly, about 3 to 4 minutes. Finish by garnishing with fresh basil and serve immediately.