Place potatoes in a large pot and cover with cold water by 2 inches. Salt generously (about 1 tablespoon). Bring to a boil over high heat, then reduce to a simmer and cook until just tender when pierced with a knife, 12 to 15 minutes. Drain and spread on a baking sheet to cool slightly, about 10 minutes.

While potatoes cool, bring a fresh pot of salted water to a boil. Add green beans and cook until bright green and crisp-tender, 3 to 4 minutes. Drain and immediately plunge into ice water to stop cooking. Drain well and pat dry.

In a small jar or bowl, combine olive oil, red wine vinegar, Dijon mustard, lemon zest, lemon juice, salt, and pepper. Shake or whisk until emulsified.

Transfer warm (not hot) potatoes to a large mixing bowl. Add shallots, capers, and half the parsley and chives. Pour dressing over and gently fold to coat. The warm potatoes will absorb more flavor.

Fold in green beans. Taste and adjust seasoning with more salt, pepper, or vinegar as needed. Scatter remaining herbs over top. Serve at room temperature or chilled.