Sauté onions in butter. I recommend using half of a tablespoon of butter to begin with. If that isn't enough add the other half.
I made a double batch so I have a full pound of frozen broccoli.
I boiled the broccoli for 5 minutes to thaw it a bit. This helps extract the water from the freezing process.
Onions are done, enough. I wish I had used less butter.
Melt the 4 tablespoons of butter in your pot.
Add your flour and whisk immediately and keep whisking for 3 minutes.
While whisking, pour in your half and half. You don't want it to curdle so keep whisking. Whisk for 3-5 minutes.
The soup is ready to simmer for 20-25 minutes over low-medium heat.
After your base has simmered for 20-25 minutes, add your carrots and broccoli. Let the soup simmer for another 20 minutes or until the produce is tender. Stir occasionally. .
After the soup is done simmering, add your cheese and stir right away. You'll want to lose the whisk and use a spoon by now. Stir until melted; about 5-10 minutes.
Add salt and pepper to taste. I liked the combination of white and black pepper and sea salt.
Just add sourdough bread and you're set. I found that there was room for a little less butter but overall it was very good.