Place the water in a large pot and bring it to a boil. Add the chicken stock, onion, carrot and chicken fillets. Season with salt and pepper and simmer for 20 minutes.
Meanwhile, if you're using fresh asparagus, boil it in a pot of salted water for 15-20 minutes until tender. Cut it into small chunks.
Once cooked, transfer the chicken to a dish and allow to cool. Once it has become cool enough to handle without burning yourself, dice it into small pieces.
Strain the chicken soup and reserve for later. Discard the vegetables.
For the white sauce:
Melt the butter in a saucepan and stir in the flour, thoroughly to form a roux. Slowly add the milk, little by little, then 4/5 cup of the chicken soup, mixing until all the flour dissolves. Adjust the seasoning with salt and pepper if necessary and allow it to cook and thicken for another 10 minutes or so, making sure to stir every so often so it doesn't burn.
Throw in the chicken and asparagus pieces and heat everything through.
Warm the tartlet cups according to package instructions and fill them with the chicken and asparagus mixture.
Sprinkle a pinch of chopped fresh parsley or dill over the top and serve.