In an Instant Pot, heat the olive oil on the sauté setting. Add the diced onion and cook for about 3-4 minutes until softened. Stir in the minced garlic, oregano, basil, and red pepper flakes, cooking for another minute until fragrant.
Season the chicken breasts with salt and pepper, then add them to the pot. Pour in the chicken broth and diced tomatoes. Close the lid and set the Instant Pot to high pressure for 10 minutes.
Once the cooking time is complete, allow the pressure to release naturally for 5 minutes, then perform a quick release for any remaining pressure.
Remove the chicken breasts and shred them using two forks. Return the shredded chicken to the pot.
Stir in the heavy cream, chopped spinach, and grated Parmesan cheese. Cook on the sauté setting for an additional 3-5 minutes until the spinach is wilted and the soup is heated through.
Taste and adjust seasoning with additional salt and pepper if needed. Serve hot.