Bring water and onion to a boil in a large soup pot. Add seasonings and half of the broccoli. Bring to a boil again. Add soup base and lower heat. In a separate saucepan, make a roux. When the roux thickens, gradually stir into the soup, whipping with a wire whisk to avoid lumps. Heat milk and cheese to just below boiling point until the cheese melts, stirring constantly. Blend into the soup and add the remaining broccoli. Just before serving, add beer. Mix well. Makes 2-1/2 quarts.