Place mixed vegetables and frozen broccoli into a large soup pot. Pour in chicken stock and bring to a boil; reduce heat to medium-low and simmer until vegetables are tender, about 20 minutes.

Meanwhile, melt butter in a saucepan over medium heat; stir in flour until smooth and cook for a few minutes until lighter in color. Whisk in milk and simmer until thickened, about 10 minutes. Stir in cheese until melted; stir cheese mixture into vegetables. Cook and stir until vegetables are tender, about 5 minutes more.