Heat the oil and butter in a pot over medium heat. Sauté the onions and garlic for about 3 minutes. Stir in the flour and cook for another 1-2 minutes.
Stir the broth and broccoli florets into the pot and simmer for about 15 minutes. Season with salt and pepper. Taste to check the flavor; if the flavor is weak, adjust the seasoning and simmer for 5-10 minutes more to concentrate the flavor. Add the cream and simmer for another 2-3 minutes.
Soup should be nice and thick (if the soup is too thin, dissolve a little more flour and an equal amount of water, then add it to the simmering soup until you reach a nice, thick consistency).
Once it is at the desired consistency, remove from heat and stir in the cheese.
Season well with salt and pepper to taste. Serve with crackers.