Preheat the oven to 180°C/Gas 4 (for the coddled eggs). Slice the cherry tomatoes in half, place in a bowl and season with salt and pepper. Top with the basil leaves, lemon zest and a little drizzle of olive oil. Stir together and set aside to marinate.
Add the potatoes to a pan of boiling salted water and cook for 12–15 minutes, or until just tender; remove with a slotted spoon and set aside. Add the green beans to the boiling water and blanch for a minute, then drain.
For the coddled eggs, generously butter 4–6 espresso cups or small ramekins and sprinkle with salt and pepper. Lay a basil leaf and an anchovy fillet in the base of each cup or ramekin, then crack an egg in. Stand them in a deep baking tray and surround with enough boiling water to come three-quarters of the way up the sides. Carefully transfer to the oven and cook for 6-8 minutes, or until the whites are set but the yolks are still creamy.
Meanwhile, heat a wide frying pan until it is very hot and then add 2 tbsp olive oil. Add the potatoes, cut side down, and swirl the oil around them. Once the potatoes are golden brown on the flat side, turn them over and add the blanched green beans. Season with salt and pepper and cook, tossing the vegetables, for a further 30 seconds, then take off the heat.
To assemble the salad, tear the outer leaves from the lettuce hearts and arrange on a platter or individual plates. Cut the hearts into wedges and dip the leafy ends in tarragon mayonnaise. Arrange on the platter (or plates) with the sautéed potatoes, green beans, marinated tomatoes and olives.
Run a knife around the coddled eggs, then lift them out and place on the salad. Break the poached salmon into large flakes and arrange on top. Drizzle over a little more tarragon mayonnaise and grind over some pepper. Serve at once, with the rest of the mayonnaise in a bowl on the side.