Set the Instant Pot on the sauté setting and heat olive oil and melt the butter.
Add the chopped bacon and cook until cooked till done.
Mix in the onion, carrots and celery and sauté until the onion is softened.
Add in the garlic, Italian seasoning, thyme and rosemary sprig and cook for about 30 seconds until fragrant.
Place the chicken breasts in and pour in the stock, scraping to make sure nothing is stuck on.
Close the lid, seal the valve, set to Pressure Cook (or Manual based on the model of your Instant Pot) and set on high pressure for 12 minutes.
Once done, perform a quick pressure release (QPR), open the lid, and transfer the chicken to a cutting board. Shred the chicken using two forks.
Set the Instant Pot on the sauté setting again. Add shredded chicken back in, followed by the spinach and gnocchi, stirring to incorporate. Cook for 3 minutes. Add the heavy cream and Parmesan cheese and cook for 1 more minute.