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Peel and roughly chop the onion and garlic cloves. This doesn't need to be beautiful, cause we're going to blend the soup anyway. Roughly chop the broccoli, if you're also using the stem you need to peel it first.
Slice the smoked chicken in thin slices and set aside.
Heat up a bit of oil in one of the medium-sized pans and sauté the onion and garlic. Add the broccoli and cook it a few minutes more. Pour in the chicken stock and bring to a boil.
Let it cook on low heat for about 10 minutes, depending on the size of the broccoli pieces. Meanwhile heat butter in the second pan and add the flour all at once to create a roux.
Cook on low heat for a few minutes. Pour in the milk and use a whisk to create a smooth mixture. Puree the broccoli until it's smooth. Stir the grated cheese into the milk mixture and pour in a bit of the broccoli mixture.
Keep on stirring with the whisk until all of the broccoli is added and the soup is smooth. Season with pepper and salt and serve with the smoked chicken.
Garnish with parsley if you like and serve with some homemade bread. Enjoy!