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Heat a large pot of water for the vegetables. Toss the onion with 2 TBSP of the vinegar and set in the refrigerator.
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Wash the potatoes, place in small saucepan, cover with cold water, add 1 tsp salt, and bring to a boil. Simmer until tender when pierced with a knife, about 25 minutes, then drain. Cut the stem ends off the beans, along with the tails if they’re tough. If the carrot are small and tender, you don’t need to peel them. Leave them whole or halve lengthwise with about an inch of the stems. Wash and dry the lettuces and herbs.
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In a bowl mash the garlic with ½ tsp salt. Whisk in the mustard, the remaining 2 TBSP vinegar, and the oil, making a thick, emulsified dressing.
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When the water boils, season well with salt, then add the beans and cook until tender, 4 to 8 minutes. Scoop them out and put them on a towel to dry briefly, toss them while still hot with half the dressing. Season with salt and pepper and heap them in the center of a platter. Boil the carrots until tender-firm, 4 to 6 minutes, then drain and dress lightly.
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Arrange the lettuces on the platter. Place the chicken at either end, breaking it up slightly. Halve the potatoes and arrange them on the platter. Spoon the remaining dressing over the lettuce and potatoes and scatter capers over all, along with the onions, drained of their vinegar. Tuck in the radishes and carrots, add the purslane and herb sprigs. Season everything with pepper. Present the salad arranged. Toss it before serving.