Place the broccoli, carrots, onion, and garlic in the bottom of 5-to-6-quart slow cooker. Top with the cream cheese, oregano, nutmeg, and chicken broth. Stir to evenly distribute the ingredients. Cover and cook on high for 2 hours or low for 4-6 hours, until the broccoli is tender.
Remove the lid and stir in the evaporated milk. With an immersion blender, puree about 3/4 of the soup, so that it is mostly smooth but still has a few chunks of broccoli remaining. Alternatively, you can puree the soup in 1 cup-sized batches in a food processor or blender. (Be careful—it's hot and prone to splatter!)
Recover the soup and let cook on low for 10 minutes until warmed through. Stir the salt, pepper, and grated cheese. Taste and add additional salt and pepper as desired. Recover and cook on high until the cheese is fully melted and the soup is hot, about 5 additional minutes. Serve warm.
The type of cheese you use is very important to this recipe. I highly advise grating your own from the block, as many brands of pre-grated, bagged cheese are coated with starch to prevent sticking. While convenient, this starch can prevent the cheese from melting smoothly. I also recommend using regular, not light cheese for this reason—reduced fat cheese sometimes does not melt as well.