Prepare an ice water bath in a large bowl (cold water and 8-10 ice cubes) near the sink. In a large sauce pan or stock pot, blanch broccoli florets in boiling water 3-4 minutes until a fork can just slide through the broccoli stems. Then drain and immediately submerge the broccoli into the ice water. In a separate stock pot at medium heat, melt the butter. Then, saute the onions for 5-7 minutes until soft. Add the flour, whisking it for 1-2 minutes to incorporate fully into the butter to form a roux. Now add the chicken broth and half & half whisking it at medium high temperature until the mixture is brought to a boil. Reduce the temperature to medium and whisk in the cheddar cheese. Add the salt, pepper and vermouth and taste. Then add a few dops of hot sauce if you like and taste again. Now drain the ice water bath and add the florets into the cheddar cheese soup and serve right away in soup bowls.