1 tsp. Dijon mustard
1 egg yolk
1 cup canola oil, plus more for frying
2 tsp. white wine vinegar
2 tsp. fresh lemon juice
1/2 tsp. kosher salt, plus more to taste
Freshly ground black pepper, to taste
2 lb. Yukon Gold potatoes, peeled and cut into 1/4″-thick sticks
2 1/2 lb. mussels, debearded and scrubbed
2/3 cup dry white wine
2 tbsp. unsalted butter, cubed
3 ribs celery, finely chopped
1 1/2 leeks, light green and white parts, cut into 1/4″-thick slices
1/2 large yellow onion, finely chopped
Make the mayonnaise: In a large bowl, whisk mustard and egg yolk. Whisking constantly, slowly drizzle in oil in a thin stream until it begins to emulsify; whisk in vinegar, lemon juice, salt, and pepper. Set aside.
Make the fries: Pour oil into a 6-qt. Dutch oven to a depth of 2″, and heat over medium-high heat until a deep-fry thermometer reads 375°. Add potatoes and cook until tender, about 8 minutes. Using a slotted spoon, transfer fries to a rack set over a rimmed baking sheet; chill.
Increase oil temperature to 385°. Working in batches, add chilled potatoes and cook until golden brown and crisp, about 4 minutes. Using a slotted spoon, return fries to rack; season with salt.
Meanwhile, make the mussels: Heat a 12″, high-sided skillet over high heat. Add mussels, wine, butter, celery, leeks, and onions; season with salt and pepper, and cover skillet. Cook, occasionally shaking skillet, until all mussels are opened, about 5 minutes. Divide mussels between 2 large bowls. Serve with fries and mayonnaise.
To take the flavor of moules frites in different directions, omit the wine and add one of the following ingredients to the pan with the mussels, butter, celery, leeks, and onion: 1 tablespoon blanched, minced garlic and 3/4 cup heavy cream; 1 tablespoon curry powder; 1 tablespoon piment d’Espelette; 1 tbsp. Pernod; 2/3 cup wheat beer.