Melt the butter over medium heat in a heavy-bottomed Dutch oven. Add the onion and garlic, stirring until soft and translucent—about 4 minutes.

Sprinkle the flour over the onion mixture, stirring constantly to form a smooth paste, or roux. Cook for 1–2 minutes until it smells slightly nutty—this removes the raw flour taste.

Gradually whisk in the chicken broth, making sure no lumps remain. Bring to a gentle simmer, whisking every so often until the liquid thickens slightly, about 3 minutes. Pour in the milk and Dijon mustard, then add the broccoli florets.

Cover and reduce heat to low. Let it simmer for 10–12 minutes, until the broccoli is tender when pierced with a fork. Use an immersion blender to partially puree the soup—leave some chunks for texture. If you don’t have one, transfer half the soup to a blender and pulse carefully.

Stir in the shredded cheddar cheese, one handful at a time, until fully melted and silky. Season with salt, pepper, and nutmeg. Taste and adjust seasoning. Serve hot, with garnishes if desired.