Heat 3 tablespoons of olive oil in a large pot over medium heat. Once shimmering, add the diced onion and cook for 5 minutes until softened and translucent, stirring occasionally. Add the freshly minced garlic and cook for 1 minute until fragrant. Pour in the crushed tomatoes along with 1½ tablespoons Italian seasoning, 1 teaspoon sugar, ¾ teaspoon salt, and ¼ teaspoon red pepper flakes. Stir well to combine, then reduce heat to low and let simmer for 20-25 minutes, stirring occasionally. I like using San Marzano tomatoes because they have a natural sweetness that requires less sugar to balance the acidity.
While the tomato sauce simmers, bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente, then drain and set aside. In a separate large skillet, heat 1½ tablespoons of olive oil over medium-high heat. Season the 1-pound chicken breast pieces with salt and black pepper, then add them to the hot skillet in a single layer. Cook for 7-10 minutes, stirring occasionally, until the chicken is golden brown and cooked through (internal temperature of 165°F). Remove the cooked chicken from the skillet and set aside on a plate.
Pour the finished tomato sauce from Step 1 into the same skillet where you cooked the chicken (the browned bits add great flavor). Reduce heat to low and stir in the 1 cup of heavy cream, mixing gently until the sauce reaches a beautiful pink color. Simmer for 2-3 minutes until the sauce is hot and well combined. Add the ¾ cup of freshly grated Parmesan cheese, stirring constantly until completely melted and the sauce is smooth and creamy. I always grate my Parmesan fresh because pre-grated versions contain anti-caking agents that prevent smooth melting.
Add the cooked chicken from Step 2 back into the cream sauce, stirring gently to coat evenly with the pink sauce. Add the drained pasta to the skillet and toss everything together until the pasta is well coated and heated through, about 1-2 minutes. Divide into bowls and garnish generously with fresh parsley and additional Parmesan cheese if desired.