Shred the cheddar and cut broccoli into small florets; peel and dice tender stems if using.
Warm a large heavy pot over medium heat and melt the butter.
Cook onion and carrots in the butter, stirring, until softened and lightly golden, 5–6 minutes.
Stir in garlic and cook 30 seconds until fragrant.
Sprinkle in the flour and stir constantly for 2 minutes to form a blond roux.
Slowly whisk in the chicken broth until smooth and lump-free.
Add broccoli, Worcestershire, dry mustard, salt, pepper, and nutmeg. Simmer gently until broccoli is tender, 12–15 minutes.
For a creamier texture, briefly pulse with an immersion blender, leaving some small pieces.
Reduce heat to low. Stir in the half-and-half and warm gently without boiling, 3–4 minutes.
Remove from heat. Add cheddar by handfuls, stirring until melted and silky. Taste and adjust seasoning; thin with a splash of broth if needed.