Place the quartered potatoes in a large pot and cover with cold, salted water. Bring to a boil and cook until tender when pierced with a fork, about 15-20 minutes. Drain well.

While potatoes are cooking, blanch the green beans in boiling salted water for 2-3 minutes until crisp-tender. Immediately transfer them to an ice bath to stop the cooking process, then drain thoroughly.

Prepare the vinaigrette: In a small bowl, whisk together the Dijon mustard, red wine vinegar, sugar, salt, and pepper. Slowly drizzle in the olive oil while continuously whisking until the dressing is emulsified.

While the potatoes are still warm (this helps absorption), place them in a large bowl. Pour about half of the vinaigrette over the potatoes and gently toss to coat. Allow them to rest for 5 minutes.

Add the blanched green beans, minced shallots, and chopped parsley to the potatoes. Pour the remaining vinaigrette over the mixture and gently fold everything together. If desired, fold in the chopped hard-boiled eggs.

Taste and adjust seasoning if necessary. Serve warm or at room temperature. This salad benefits from sitting for 15-30 minutes before serving to allow the flavors to meld.