Turn the instant pot to sauté and add butter or olive oil. Once melted, add diced onion and carrots and cook until softened. Stir in minced garlic, salt, dried rosemary, and black pepper.
Pour in chicken broth, milk, and heavy cream. Stir well, then add the wild rice and chicken thighs.
Secure lid and ensure the steam valve is in the sealing position. Set to cook on high pressure for 25 minutes.
When the cook time ends, allow the instant pot to naturally release for 10 minutes, then carefully release any remaining pressure.
Remove chicken and shred using two forks. Return the shredded chicken to the pot.
Turn on sauté again, add spinach, and stir until wilted.
Ladle soup into bowls and top with grated parmesan and a squeeze of lemon juice before serving.