In a large pot, melt 1/2 cup of butter over medium heat.
Add the chopped onion.
Cook for 3 to 4 minutes until the onion is softened.
Sprinkle 1/4 cup of flour over the onions.
Stir to combine and cook for 1 minute.
Gradually pour in 2 cups of milk and 2 cups of half-and-half.
Stir constantly to prevent lumps.
Add a dash of nutmeg, the broccoli florets, salt, and plenty of black pepper.
Cook for 20 minutes until the broccoli is tender.
Use an immersion blender to puree the soup, leaving some texture.
Add 2 cups of grated cheddar cheese.
Stir until melted.
Serve hot and enjoy!