Set your Instant Pot to the sauté function. Add the olive oil and let it heat up. Once hot, toss in the diced onion, minced garlic, diced carrots, and diced celery. Sauté these vegetables for about 5.
Season the chicken breasts with dried thyme, oregano, salt, and black pepper. Place the seasoned chicken into the pot with the sautéed vegetables. Pour in the chicken broth, ensuring the chicken is submerged. Close the lid and set the valve to sealing. Cook on high pressure for 10.
Carefully perform a quick release of the pressure. Remove the chicken from the pot and shred it using two forks. Return the shredded chicken back into the pot. Add the potato gnocchi to the mixture. Stir well to combine everything. Set the Instant Pot to sauté mode again and cook for about 3-5.
Pour in the heavy cream and stir well. Allow it to heat through for a couple of minutes.
Stir in the fresh spinach and grated Parmesan cheese. Taste your soup and adjust the seasoning if needed. Once everything is well combined and heated through, serve!