Restaurant Hubert's le grand aïoli

Ingredients

• 400 gm chat potatoes, scrubbed
• 400 gm Dutch carrots, peeled
• 360 gm baby fennel bulbs, quartered lengthways
• 400 gm broccolini
• 200 gm mussels, scrubbed and de-bearded
• 100 gm red witlof leaves
• 100 gm white witlof leaves
• 4 baby cucumbers, halved lengthways
• 300 gm radishes, halved
• Aïoli
• 6 garlic cloves (about 25gm), plus 1 clove extra
• 3 soft-boiled eggs
• 2 egg yolks
• 25 gm Dijon mustard (1¼ tbsp)
• ¼ tsp ground piment d’Espelette, plus extra to serve (see note)
• 250 ml canola oil (1 cup)
• 125 ml extra-virgin olive oil (½ cup)
• 2 tsp lemon juice, or to taste