Set the Instant Pot on the SAUTE setting. Add the olive oil and heat.
Add the chopped bacon and cook until cooked thoroughly, around 5 minutes.
Add the onions and carrots and cook until the onion is slightly translucent, around 5 minutes more.
Add the garlic, rosemary sprig, thyme and Italian seasoning and cook until fragrant, around 1 minute.
Pour in a little bit of the stock and scrape any stuck food particles that are sticking to the bottom of the Instant Pot insert. NOTE: this will prevent a BURN warning.
Add the chicken breast and rest of the stock.
Close the lid and valve and set on high pressure for 10 minutes.
Allow a NATURAL RELEASE for 5 minutes. Then perform a QUICK RELEASE to release any remaining pressure.
Remove the chicken breast and rosemary sprig.
Select the SAUTE setting and add the gnocchi. Cook until all the gnocchi have floated to the top of the soup (2-3 minutes).
While cooking the gnocchi use two forks to shred the chicken breast.
After the gnocchi is done cooking turn off the Instant Pot. Stir in the cheese and spinach. Stir for 1 minute until the spinach begins to wilt.
Add the shredded chicken and stir in the heavy cream. Serve hot!