Combine the wild rice, chicken, carrots, celery, shallots, chicken broth, thyme, sage, crushed red pepper flakes, and a pinch of salt and pepper in the slow cooker. Add 1/2 cup of water. Cover and cook on low for 5–6 hours or on high for 2–3 hours.
Shred the chicken using two forks, then stir in the heavy cream, Parmesan cheese, and spinach. Let it heat through for about 30 minutes. Add parsley at the end. Adjust the soup's consistency with an extra 1/2 cup of water if too thick.
Preheat the oven to 425°F. On a baking sheet, mix the mushrooms, melted butter, smashed garlic, rosemary, lemon zest, and a pinch of salt and pepper. Roast for 25–30 minutes, stirring midway, until golden and crisp.
Mash the roasted garlic with a fork, and stir it into the soup along with the roasted mushrooms and any remaining butter.
Serve in bowls topped with rosemary sprigs and additional Parmesan cheese.