Prepare the Hard-Boiled Eggs: Place eggs in a saucepan, cover with cold water, bring to a boil over medium-high heat, remove from heat, cover, and let sit for 10 minutes. Transfer eggs to cold water to cool, then peel under running water.

Cook the Potatoes: Wash and halve or quarter the new potatoes. Place in a pot, cover with salted water, and bring to a gentle boil. Cook until tender but firm, about 12 minutes.

Add the Green Beans: Stir in the trimmed green beans once potatoes are cooked. Boil for an additional 2 to 3 minutes until beans are bright green and tender-crisp.

Ice Bath: Drain potatoes and green beans in a colander, then plunge them into ice water for 5 minutes to stop cooking. Drain again and set aside.

Make the Vinaigrette: In a jar, combine olive oil, lemon juice, wine vinegar, garlic, mustard, capers, salt, and pepper. Shake vigorously until emulsified.

Assemble the Salad: In a bowl, combine cooled potatoes and green beans with parsley and chives. Drizzle with half vinaigrette and toss. Slice hard-boiled eggs and arrange on top, drizzle with remaining vinaigrette.

Serve and Enjoy: Let salad rest for at least 30 minutes, or refrigerate for a few hours. Serve chilled or at room temperature.