Heat oil in a large pot or skillet over medium-high heat. Add chicken and cook until browned on all sides. Remove chicken from the pan and set aside.
Add onion, carrot, garlic, Italian seasoning, and chili flakes to the pot. Cook until the onions soften, about 3 minutes.
Add the bell peppers, crushed tomatoes, balsamic vinegar, salt, and pepper. Bring to a simmer and cook for about 10 minutes or until the sauce thickens slightly.
Add chickpea pasta and chicken broth to the pot. Bring to a boil, then reduce heat to low, cover, and simmer according to pasta package directions. It should take 4-5 minutes.
Once the pasta is almost cooked, add in heavy cream and Parmesan cheese. Return the browned chicken to the pot and mix well. Let it simmer for 2-3 minutes or until the pasta and chicken are fully cooked.
Add spinach and cook until wilted. Turn off the heat and serve.
Notes
For the best flavor, make sure to sear the chicken pieces. This adds depth to the overall taste.
Don't overcook the chicken, as it can become dry. Aim for an internal temperature of 165 F.
For an extra-rich sauce, use heavy cream and high-quality Parmesan cheese. If you prefer a lighter option, try half-and-half or Greek yogurt.