Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Pour in oil and swirl to coat the pot. Add onion, carrot, and celery. Cook until onion is soft and translucent, about 5 minutes. Stir in garlic, parsley, rosemary, thyme, oregano, and red pepper flakes; cook for 1 minute. Pour in chicken broth and stir to combine. Season with salt and pepper. Place chicken breasts into the liquid.

Close and lock the lid with the vent closed. Select high pressure according to manufacturer's instructions; set timer for 9 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Release remaining pressure naturally.

Unlock and remove the lid. Select Saute function. Add gnocchi and spinach and cook until gnocchi float to the top, 2 to 4 minutes. Remove the chicken to a cutting board and coarsely chop; return to the soup. Taste soup and adjust salt if necessary.