In the bowl of your instant pot, combine the rice, chicken, carrots, celery, shallots, broth, thyme, sage, crushed red pepper, and a pinch each of salt and pepper. Cover and cook on high pressure for 15 minutes. Quick release the steam.
Switch the instant pot to sauté. Shred the chicken with 2 forks. Stir in the cream, parmesan, and spinach. Cook until warmed through, about 10 minutes. Stir in the parsley.
To make the mushrooms. Preheat the oven to 425 degrees F. On a baking sheet, combine the butter, mushrooms, garlic, rosemary, lemon zest, and a pinch each of salt and pepper. Transfer to the oven and roast for 25-30 minutes, stirring halfway through cooking, until the mushrooms are golden and crisp.
Mash the garlic with a fork, then stir in the mushrooms, mashed garlic, and any butter left into the soup.
Divide the soup among bowls and top with rosemary and additional parmesan.
