In a large heavy bottom pot (I used a dutch oven) melt the butter over medium heat. Drop in the shallots and sauté until softened, but not browned...about 3 minutes.

Whisk the flour into the melted butter and shallot mixture. Whisk until the flour becomes lightly brown...about 2 minutes.

Whisk in the milk and the vegetable stock. Stir until the flour has dissolved.

Pour in the broccoli and bring to a boil. Once boiling drop down to a simmer and let the soup simmer for about 20 minutes, or until the broccoli stalks are soft.

Use an emersion blender, or carefully transfer the broccoli soup to a blender. Whiz up the soup until it has reached the desired smoothness. Some people like their soup a little chunky. My kids like the soup totally smooth. I kept a few pieces of broccoli aside and added the pieces to my bowl of soup, as I like my soup a little chunky.

Next, stir in the cheddar cheese into the warm soup and continue stirring until the cheese has completely melted.

Taste the soup and add salt and pepper to taste. At this point you may want to add more stock, if your soup is thicker than you like.

Next, stir in the cheddar cheese into the warm soup and continue stirring until the cheese has completely melted.