Heat the olive oil in a Dutch oven or pot over medium-high heat. Add in the onion and sauté for 2-3 minutes. Stir in the garlic, Italian seasoning, oregano, red pepper flakes, salt and pepper and cook another minute.
Stir in the tomato paste, cooking it for a minute. Add in the sun-dried tomatoes and parmesan cheese rind, then pour in the broth. Season with additional salt and stir everything together, scraping up any brown bits from the bottom of the pot.
Bring to a boil, then pour in the uncooked pasta. Lower the heat to medium and cook until the pasta is al dente. You'll want the pasta to still be semi-hard, as it will continue to cook in the hot broth.
Add in the chicken, spinach and parmesan cheese. Stir everything together and cook until the spinach is wilted, 1-2 minutes.
Remove the soup from the heat and stir in the heavy cream. Add in the Greek yogurt and stir until it is thoroughly incorporated. Taste for seasoning and add additional salt if needed. Serve immediately.