For the Aioli:
1 cup canola oil
½ cup olive oil
2 egg yolks
2 tablespoons water
2 teaspoons Dijon mustard
1 teaspoon fresh lemon juice
1 large garlic clove, smashed into a paste
¾ teaspoon kosher salt
For the French Fries:
2 pounds large russet potatoes, rinsed
Canola oil, for frying
Kosher salt, to taste
1 tablespoon finely chopped flat-leaf parsley
For the Mussels:
3 tablespoons olive oil
2 medium leeks (light green and white parts)—cut into ½-inch rounds, rinsed and patted dry
½ cup finely diced fennel (½ fennel head)
¼ cup finely diced shallot (1 medium shallot)
¼ cup finely diced celery (1 celery stalk)
Kosher salt, to taste
5 garlic cloves, thinly sliced
4 thyme sprigs
1 bay leaf
Pinch crushed red pepper flakes or piment d'Espelette
3 cups dry white wine
4 pounds mussels, debearded and scrubbed
¼ cup finely chopped flat-leaf parsley
Warm crusty bread, for serving
. Make the aioli: In a measuring cup, combine both oils. In a large mixing bowl, whisk together the egg yolks, water, mustard, lemon juice, garlic paste and salt. Slowly add the oil, drop by drop, in a thin, steady stream until it begins to emulsify. Continue adding the olive oil, whisking constantly, until all the olive oil is incorporated. Make ahead: The aioli can be made up to 3 days ahead. Keep it covered and chilled until ready to use.
2. Make the french fries: Using a mandoline, slice the potatoes lengthwise very thinly (about 1/16-inch thick), making sure to use even pressure when slicing. Stack the potato shavings on top of one another, then slice the potatoes lengthwise into ¼-inch matchsticks. Transfer the potato slices to a large bowl. Cover with cold water and slice the remaining potatoes into matchsticks.
3. Using your hands, gently stir the potatoes until the water becomes cloudy; drain and repeat the process 4 to 5 times, until the water is clear. Set aside.
4. Working in batches, transfer the potatoes to a salad spinner and spin until all of the water is spun out.
5. Meanwhile, make the mussels: In a large heavy-bottomed skillet or, preferably, a high-sided cast-iron pan over medium heat, add the olive oil. When the oil is warm, add the leeks, fennel, shallots and celery. Season with salt and cook, stirring often, until soft but not brown, 8 to 10 minutes. Add the garlic, thyme, bay leaf and red pepper flakes, and cook for 2 minutes more. Increase the heat to high and add the wine. Bring to a boil and reduce by half, 2 minutes. Add the mussels and cook, covered, shaking the pan often and checking every 30 seconds until a few mussels are open. As the mussels open, remove them using a slotted spoon and transfer to a bowl. Discard any mussels that have not opened. Place the pan's lid over the bowl to keep the mussels warm and remove the pan from the heat.
In a medium mixing bowl, temper ⅓ cup of the prepared aioli by whisking it with ½ cup of the warm mussel cooking liquid. Taste and adjust the seasoning if necessary. Return the liquid to the skillet. Set the mussels and vegetables aside.
7. While the mussels are cooking, transfer the potatoes to a large, heavy-bottomed pot, spreading them in one even layer. Pour in enough oil to cover the potatoes by 1-inch. Place the pot over medium heat and cook, loosening the potatoes with a spatula, until the potatoes are tender, 16 to 18 minutes.
8. Increase the potatoes to medium-high heat (the oil will begin to bubble rapidly) and fry the potatoes until crisp and golden, 6 to 8 minutes. Using a slotted spoon, transfer the fries to a paper towel-lined baking sheet and immediately season them with salt while still hot. Sprinkle with parsley.