Take a pan and melt 1 tablespoon butter over medium heat. Now add onion and saute it for 5 minutes.
Whisk ½ cup melted butter and flour together on a medium heat. Cook until the flour loses its granular texture. You can add 1 to 2 tablespoons of milk if required to keep the flour from burning, cook for 3 minutes
Now add milk to the mixture while whisking constantly. Now stir chicken stock into milk mixture and cook for about 20 minutes until the mixture thickens.
Now add all the vegetables and cook until vegetable are tender, for about 20 minutes.
Now add cheddar cheese into the vegetable mixture and cook till the cheese melts.
Cheddar broccoli soup is ready! Season with salt and pepper and serve it hot.