Prepare Hard-Boiled Eggs: Place the eggs in a saucepan and cover with cold water. Bring to a boil, remove from heat, cover, and let sit for 10 minutes. Cool under running water and set aside.

Cook Potatoes: Halve the new potatoes and place in a large pot of salted water. Bring to a boil and cook for 12 minutes. Add green beans for the last 2 minutes, then drain.

Shock Vegetables: Prepare an ice-water bath. Transfer drained potatoes and green beans into the ice-water bath for 5 minutes.

Make Vinaigrette: In a jar, combine olive oil, lemon juice, white wine vinegar, minced garlic, Dijon mustard, capers, and black olives. Shake until emulsified.

Combine Salad: In a bowl, combine parsley and chives with half of the vinaigrette. Toss in cooled potatoes and green beans. Top with quartered eggs and drizzle with remaining vinaigrette.