In a large dutch oven over medium heat, add olive oil, minced garlic and onion; cook for about 4 minutes, stirring frequently, until fragrant.
Add butter and flour, whisking constantly, about 3 minutes. While whisking, slowly pour in vegetable broth, then whisk in milk.
Add broccoli, carrots, celery, salt, black pepper, onion powder, garlic powder, and nutmeg. Bring mixture to a boil, then reduce heat to simmer for about 15-20 minutes, stirring occasionally, until broccoli and vegetables are soft.
Add in half and half and shredded cheese, stirring until melted.
Optionally, blend part of soup (about 2 cups worth) using an immersion blender or high-power blender. Stir back into the pot and enjoy!