Prepare the Vegetables and Cheese: Place the broccoli, carrots, celery, onion, and garlic in the bottom of a 5-to-6-quart slow cooker.
Add Spices and Liquids: Top with the cream cheese, oregano, nutmeg, paprika, thyme, and chicken broth. Stir to evenly distribute the ingredients.
Slow Cooking: Cover and cook on high for 2 hours or low for 4-6 hours, until the broccoli is tender.
Blending the Soup: Remove the lid and stir in the evaporated milk. With an immersion blender, puree about ¾ of the soup, so it is mostly smooth but still has a few chunks of broccoli remaining. You can also puree the soup in 1 cup-sized batches in a food processor or blender. (Be careful—it’s hot and prone to splatter!)
Final Touches: Recover the soup and let cook on low for 10 minutes until warmed through. Stir in the salt, pepper, cayenne pepper (if using), and grated cheese. Taste and adjust seasoning as desired.
Melting the Cheese: Recover and cook on high until the cheese is fully melted and the soup is hot, about 5 additional minutes.
Serving: Serve warm, garnished with a dollop of sour cream and a sprinkle of fresh parsley.
