In a large pot, sautee butter, onions and carrots until the onions are soft, about 5-7 minutes.
¼ cup butter, 1 small onion, 2 carrots
Add garlic and flour. Cook for about 2 minutes, stirring so mixture won't burn. Stir in dried mustard, salt and pepper.
1 clove garlic, ¼ cup flour, ½ teaspoon salt, ½ teaspoon pepper, ½ teaspoon dried mustard
Slowly add broth while stirring. Then add broccoli. Bring to a low simmer and cook for about 10 minutes, until broccoli is softened.
4 cups broccoli, 4 cups chicken broth
Use an immersion blender and puree the soup until the the mixture is creamy and most of the broccoli is pureed. You can leave chunkier or puree until completely smooth.
Whisk in milk and bring back up to a simmer.
2 cups milk
Remove soup from heat and allow the soup to cool for just a few minutes. Slowly mix in cheese. Stir until melted.
8 ounces cheddar cheese
Serve hot along with additional shreddedc cheese and croutons.
Notes
Shred your own cheese instead of purchasing pre-shredded. Pre-shredded cheese can have an additive so the cheese won't clump together. This additive can prohibit the cheese from melting.
Make certain the broccoli and carrots are soft before pureeing
Puree as smooth as you'd like. You can leave large chunks, just a few chunks or make your soup very smooth and creamy.
Use an immersion blender or process in small batches in a regular blender