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Place potatoes in a large saucepan, cover with water and cook 15-20 minutes, Drain and set aside. Blanch the beans, peas, broccoli and carrots in boiling water. Remove to a large bowl of ice water. Drain and pat dry. Slice potatoes lengthwise ½-inch thick. In a large bowl, combine potatoes and blanched vegetables. Add red onion, scallions and olive oil, tossing lightly to coat vegetables.
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Prepare grill. Arrange potatoes and vegetables in adjustable grill basket. Grill vegetables over medium coals, rotating occasionally, 5-10 minutes for each side, until potatoes and vegetables start to brown. Place grilled potatoes and vegetables in a large bowl, drizzle salad dressing over all, tossing to coat evenly.
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Line 6 plates with redleaf lettuce leaves. Divide potatoes and vegetables among plates. Garnish each with a sprig of arugula.