Boil the potatoes in well- salted water until tender, about 15-20 minutes. Add the green beans to the same pan for the last 4-5 minutes of the cooking time. Remove from the heat and drain well.
Put the eggs in a saucepan and cover with cold water. Bring to the boil and simmer for 6 minutes then drain. Run the eggs under cold water until cool enough to peel.
Put the thinly sliced shallots in a small bowl and cover with boiling water from the kettle. Leave for 5 minutes, then drain, add a pinch of salt and a splash of red-wine vinegar, mix well and put to one side.
In a large bowl, mix together the red-wine vinegar, Dijon mustard and olive oil. Stir well, then add the tomatoes and capers, mixing well. Add the anchovies and tarragon and season well with salt and freshly ground black pepper to taste.
Add the cooked green beans and potatoes, the beans and the pickled shallots, mixing well to combine with the dressing in the bowl. Quarter the eggs and add just before serving.