Place the mussels in a large bowl of cold water. Use a small knife to remove any barnacles and pull away the beard. Discard any mussels that are damaged or won’t close when tapped.
Put the mussels and wine into a large sauce pan and bring to a boil shaking the pan now and then for 3-4 minutes. Discard any that have not opened. Drain into a colander. When cool enough to handle remove one side of the shell.
To make the pesto: Place the basil leaves, cheese, garlic and pine nuts in a food processor or blender. Turn on and gradually add the olive oil. Taste and season.
Heat a frying pan and add the breadcrumbs. Allow to turn golden, stirring frequently. This will take just a couple of minutes.
Preheat the grill. Place the mussels onto a grill rack and spoon a little the pesto onto each mussel then sprinkle some bread crumbs on top. Grill for 2-3 minutes to warm the pesto and give the breadcrumbs an extra crunch.