Clean mussels under cold running water. With a sharp knife remove beard (the thread like fronds) and discard together with any open mussels that don't close when tapped.
Melt butter in a large pan, add garlic and onion. Sweat for 5 minutes — don't brown.
Chop fennel, reserving feathery tops for garnish, and add to pan with mussels, wine, stock and bay leaf. Increase heat, cover with lid and cook for 5 minutes, stirring halfway through.
The mussels are cooked when all the shells are open. Discard any that remain closed. Serve in big bowls.
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