Saute onion and carrots in 1 tablespoon butter until onions are transparent. Scoop sauteed onions and carrots out and put in a bowl for later.
1 onion, 2 carrots, 1 tbsp butter
Melt ½ cup butter and flour using whisk over medium heat for 3-5 minutes.
1/2 cup butter, 1/4 cup flour
Add chicken broth and whisk to combine. Simmer (stirring occasionally) for 20 minutes.
3 cups chicken broth
Add broccoli, carrots and sauted onions and garlic. Add half and half. Cook on low for 20 minutes. Do not let it boil, but stir often.
3 cups broccoli florets, 1 clove garlic, 3 cups half-and-half
Add salt, pepper, and nutmeg.
Salt and pepper, 1/4 teaspoon nutmeg
Cook on low and add grated cheese. Stir until cheese is melted.
3 cups cheddar cheese