Add Garlic: Add the minced garlic and cook for 1 more minute until fragrant.
Make Roux: Sprinkle the flour over the vegetables. Stir constantly and cook for 2 minutes.
Add Liquids: Slowly whisk in the broth, a little at a time, to prevent lumps. Once smooth, whisk in the half-and-half.
Cook Broccoli: Bring the soup to a gentle simmer (do not let it boil). Add the chopped broccoli. Cook for 5-7 minutes, until the broccoli is fork-tender.
Blend (Optional): Use an immersion blender to pulse the soup 5-6 times to thicken the base.
The Secret Step: Turn the heat OFF. Remove the pot from the hot burner. Let it cool for 2-3 minutes.
Melt Cheese: Add the freshly shredded cheddar in small handfuls, stirring gently until each batch is melted before adding the next.
Season: Stir in the salt, pepper, and the pinch of nutmeg. Taste and serve hot.
Notes
The secret to this soup's creamy, non-grainy texture is twofold:
You must shred your own cheese from a block. Pre-shredded bags have starches that prevent smooth melting.
You must add the cheese off the heat to prevent the dairy from "breaking" or curdling.