2 tablespoons freshly squeezed lemon juice
1 tablespoon red wine vinegar
1 tablespoon whole-grain mustard
1 tablespoon Dijon mustard
1/4 cup extra-virgin olive oil
2 tablespoons walnut oil
Freshly ground black pepper
For the salad:
4 pounds mixed marble potatoes or other small potatoes
Extra-virgin olive oil, for drizzling
Sea salt and freshly ground black pepper
1 bunch scallions, white and green parts, thinly sliced
6 to 8 eggs
1 cup walnuts, toasted and coarsely chopped
Leaves from 1 bunch basil, torn
To make the vinaigrette, place the garlic on a cutting board, sprinkle with a couple of generous pinches of salt, and finely chop and smash it into a paste with the side of a chef’s knife. Whisk together the garlic paste, lemon juice, vinegar, and both mustards until smooth. Gradually whisk in the olive and walnut oils until emulsified. Taste and adjust the salt and pepper.
Heat the oven to 425 degrees. Arrange the potatoes in a single layer on two parchment-lined, rimmed baking sheets, drizzle with olive oil, and toss to evenly coat. Season with salt and pepper. Roast, shaking the sheets occasionally, until tender and brown, 40 to 45 minutes.
Transfer the potatoes to a large bowl. Toss in the scallions and the vinaigrette. Using the back of a mixing spoon, gently smash some of the potatoes just enough to break the skins. Be careful not to make mashed potatoes. Allow the dressed potatoes to sit at room temperature for 45 to 60 minutes.
About 15 minutes before serving, bring a pot of water to a boil. Lower the eggs, a few at a time, into the water and boil for 6 minutes. Remove the eggs with a slotted spoon, plunge them into an ice bath until cool enough to handle, and then peel them.
Just before serving, stir in the walnuts and basil. Arrange the salad on plates. Top each serving with a soft-boiled egg and sprinkle with salt and pepper. Serves 6 to 8.