Blend garlic, egg yolks, lemon juice, seasoning, and mustard in a food processor until smooth.
Gradually add oil while blending to create a smooth aioli paste and refrigerate.
Clean and de-beard mussels, discarding any that remain open when tapped.
Fry half the onion in a large pan with oil until soft, then add bay leaves and a splash of wine.
Add mussels and 200ml of water, cover tightly, and steam for 3 to 5 minutes until opened.
Strain the mussels over a bowl to reserve the cooking liquid and discard any unopened shells.
Remove mussel meat from shells and set aside.
Fry remaining onion, sliced leek, fennel, and carrot in olive oil until tender.
Add more wine, the reserved mussel liquid, 100ml water, saffron, and seasoning, then simmer for 10 minutes.
Whisk in a large spoonful of aioli to thicken the broth and add the mussel meat to warm through.
Place a slice of stale bread in each bowl and pour the stew over it.