Melt 1 tablespoon butter in a skillet, add onion, and fry until transparent.
Melt 1/4 cup butter in a saucepan, add flour and whisk until smooth. Then gradually pour in the milk in portions, continuing to cook over low heat and stirring constantly to avoid lumps forming.
Pour in chicken stock, stir and cook over low heat for about 15 minutes until thick.
Add broccoli, carrots, onions, and celery, stir and cook for another 20 minutes until vegetables are soft. Do not cover with a lid!
Add the grated cheese, let it melt. Season with salt and pepper to taste and serve.