Time 1 1/4 hours, plus 2 hours to chill. You'll have 1 1/4 cups extra aioli to serve with appetizer vegetables if you like (we chose radishes and blanched broccolini).
3/4 pound green beans, cut into 2-in. lengths
3 pounds small red potatoes, scrubbed
1/4 cup white wine vinegar
1/4 cup extra-virgin olive oil
1/2 medium red onion, slivered, rinsed, and drained
2 cups mayonnaise
2 tablespoons Dijon mustard
2 medium garlic cloves
About 1/2 tsp. kosher salt
Freshly ground black pepper
2 tablespoons drained capers
1 cup pitted Niçoise olives
How to Make It
Cook green beans in a large pot of boiling water just until tender-crisp, 7 to 8 minutes. Drain, cool in a bowl of ice water, drain again, and chill until ready to use.
Put potatoes in a large pot, cover with cold water, and bring to a boil over high heat. Reduce heat and simmer just until tender when pierced, about 15 minutes. Drain. When they're cool enough to handle, quarter potatoes and put them in a large bowl. Pour vinegar and olive oil over warm potatoes; add the onion, mix gently, and let stand until cool.
Meanwhile, make aioli: Finely shred peel of 1 lemon and squeeze enough lemons to get 7 tbsp. juice. In a food processor, whirl mayonnaise, lemon juice, mustard, garlic, 1/2 tsp. salt, and lots of pepper until smooth. Add lemon zest and pulse just to blend.
Mix cool potatoes with 1 cup aioli and the capers. Cover; chill until cold, along with remaining aioli.
Just before serving, gently stir in green beans, olives, more aioli if salad is too dry, and more salt and pepper to taste.