Place the mixed vegetables and frozen broccoli into a large pot. Pour in the chicken stock and bring to a boil. Reduce heat to medium-low and simmer until the vegetables are tender, about 20 minutes. Meanwhile, melt the butter in a saucepan over medium heat; stir in the flour until smooth and cook for a few minutes until the color lightens slightly. Whisk in the milk and simmer until thickened, about 10 minutes. Stir in the cheese until melted; stir the cheese mixture into the simmering vegetables. Cook and stir until the vegetables are tender, about 5 minutes more.
