Melt the butter in a large pot over medium heat, and mix in the flour. Reduce heat to low, and cook, stirring constantly, about 2 minutes. Mix in the chicken stock and milk, and season with salt and white pepper. Bring to a boil, reduce heat to low, and simmer about 10 minutes.
Bring a pot of lightly salted water to a boil. Place the broccoli in the boiling water, and cook 2 minutes, or until just tender. Remove from heat, drain, and set aside.
Gradually mix the processed cheese and Cheddar cheese into the large pot until melted. Mix in the broccoli. Continue cooking about 5 minutes.
