Add olive oil to a 3-quart soup pot on medium low heat. Add chopped onions and sauté for about 8 minutes or until translucent.
Add water and chicken stock until it comes to a low boil. Add broccoli bring to a low simmer cover and cook for an additional 10 minutes.
Once broccoli has softened puree with an emulsion blender until most of the broccoli has been pureed. --- (You can transfer to a blender to puree but, be very careful to only pulse the puree in small amounts at a time, it can blow the top off your blender and cause a big mess and potential burns.)---
Add 2 cups of milk and bring back to a low boil. In a shaker, add 1/2 cup of milk and flour shake until well blended.
While stirring constantly, slowly pour flour & milk mixture into soup until soup has reached your desired consistency
Add shredded cheddar cheese until totally melted. Once cheese is melted the soup is ready to serve. I serve with a small amount of freshly shredded cheddar cheese.