In a large soup pot melt 1/4 c. butter and saute onions until tender. Add the flour and seasonings and whisk the mixture until it cooks, about 1 minute. Slowly add the chicken broth into the onion, butter and flour mixture. Make sure to whisk continuously to avoid lumps. After the chicken broth is incorporated slowly add the milk and cook until the soup begins to thicken. Add the cheeses and steamed broccoli and simmer until the cheese melts and it is warmed through. Season with salt and pepper to taste. Serve immediately.