Melt the butter in a large pan with a tight-fitting lid. Add the chopped carrot, onion, and celery, and cook over a medium heat for 8-10 minutes until softened. Add the garlic and cook for a further 2 minutes.
Meanwhile, clean the mussels by rinsing them under the cold tap. Make sure all the shells are tightly closed and discard any that don't shut when tapped. Add the mussels to the pan with the white wine, cover, and bring to the boil. Turn down the heat and let the mussels steam away for 5-10 minutes, or until they've all opened. Discard any mussels that have remained closed.
Stir through the cream, tarragon, lemon juice and add a couple of grinds of black pepper. Heat the mussels and sauce through and serve along with some crusty bread.