In the bottom of a 5-to-6-quart slow cooker, place the chopped broccoli, diced onion, minced garlic, and grated carrots. Add the cream cheese, oregano, nutmeg, and chicken broth, stirring to combine.
Cover and cook on high for 2 hours or low for 4 to 6 hours, until the broccoli is tender and cooked through.
After cooking, remove the lid and pour in the evaporated milk. Using an immersion blender, puree roughly three-quarters of the soup, leaving some broccoli chunks for texture. If you prefer, blend the soup in small batches in a food processor or blender, being cautious with the hot liquid.
Cover again and cook on low for an additional 10 minutes until thoroughly warmed.
Stir in the salt, black pepper, and grated cheddar cheese. Taste the soup and adjust the seasoning with more salt or pepper as needed. Cover once more and cook on high for about 5 more minutes until the cheese is fully melted and the soup is heated through.
Serve hot and enjoy!