You can use the same pot of water for boiling the eggs and blanching the beans. Just watch the time.
Put the fingerling potatoes in a saucepot with enough water to cover. Bring to a boil, and add salt, and simmer, until just fork-tender.
While the potatoes are cooking, make the dressing in a serving bowl, or if you have a mortar and pestle, use that. Begin with mashing the anchovy fillets into the mustard and garlic. Add the vinegar, and slowly whisk in the oil. If too thick, add a little water. Season with salt and pepper.
Blanch the beans in salted boiling water, drain and plunge or chill down in a cold water bath. Drain the beans on a towel. Peel the eggs and cut in half. Drain the potatoes, and while still warm, drop them into the serving bowl with the dressing.
Add the beans and eggs. Gently fold all together, and serve warm in shallow white plate or bowls. This salad is also good served at room temperature.