Ingredients:

1 Tbsp salted butter
1 cup chopped yellow onion
2 garlic cloves, minced
1/2 cup grated carrot
4 Tbsp salted butter
1/4 cup all-purpose flour
2 cups half-and-half (half cream, half butter)
2 cups chicken or vegetable broth (I used chicken broth)
10 oz grated sharp cheddar
8 oz broccoli florets, chopped into small pieces
salt and pepper to taste (I used 1 tsp kosher salt and a pinch of black pepper)
1/4 tsp grated nutmeg, optional

Instructions:

1. Melt the butter in a large soup pot (6 quarts) over medium heat.
2. Once melted, add the chopped onion, garlic and grated carrot and stir occasionally until the onion has started to soften but not turn brown, about five minutes. Empty the vegetables into a medium-sized bowl and set aside.
3. Return the pot to the burner. Add the second amount of salted butter (4 Tbsp, or 2 oz). Once the butter has melted, add the flour to the pot and whisk it into the butter/onion mixture. Let the flour cook for two minutes or until it is a light blond color instead of white. (This mixture of flour and butter is called a roux. The reason you want to make a roux is so the end result is a smooth soup without flour lumps.)
4. Going slowly at first, start drizzling the half-and-half into the pot while whisking it constantly. Once the first amount of liquid has thickened, repeat three or four times the step of adding the half-and-half a little at a time and whisking until it thickens. Once the roux is noticeably thinner because of the liquid additions, you can add the rest of the half-and-half faster to the roux. Then pour in the two cups of chicken broth and stir to combine.
5. Allow the soup to come to a boil over medium heat and then turn the heat down slightly to medium-low so that the soup is still simmering. Let it cook for 10-15 minutes until it is noticeably thicker, stirring occasionally so it does not stick to the bottom of the pan and burn.
6. Whisk in the grated cheddar a handful at a time, allowing it to melt before adding the next handful. Once the cheese has melted, add in the bowl of sautéed onion, garlic and carrot to the pot. Add in the 8 oz of chopped broccoli and stir to combine.
7. Simmer the soup until the broccoli is tender but still bright green, about five minutes.
8. Add salt and pepper to taste as well as the grated nutmeg, if using.
9. Serve in a bread bowl or with a baguette.

http://tiphero.com/cheddar-broccoli-soup/