To make the pesto, in a food processor, combine the basil leaves, garlic, and nuts and pulse a few times to combine. Add the 1/2 cup olive oil and process until you have a coarse purée. (The balance of the olive oil will be added later.) Season with the salt and pepper. Set aside.

In a saucepan, combine the potatoes with water to cover, add a little salt, and bring to a boil over high heat. Reduce the heat to medium and simmer until the potatoes are cooked through but still firm. The cooking time will vary with the size of the potatoes, but very small ones will cook in about 10 minutes or so and larger ones in about 15 minutes or longer. When done, drain well and let cool to room temperature.

Meanwhile, bring a second saucepan filled with water to a boil over high beat. Lightly salt the water and then add the green beans and cook until tender-crisp, 2 to 3 minutes. Drain and refresh in a bowl filled with ice water. When cold, drain again and pat dry.

Bring a large pot of water to a boil over high heat. Salt the water liberally and add the lobsters, Cover, reduce the heat to low, and cook until they turn red, about 9 minutes. (If you have been able to find only 1-pound lobsters, cook them for about 7 minutes; if you are using 1 1/2-pound lobsters, cook for about 11 minutes, Or, you can steam the lobsters on a rack over boiling water for a particularly tender result; allow 10 minutes for 1-pound lobsters, 12 minutes for 1 1/4-pound lobsters, and 14 minutes for 1 1/2-pound lobsters.)

When the lobsters are ready, transfer them to a sink filled with ice water and leave until cold. Then, working with 1 lobster at a time, twist off the tail section from each body. Cut through the tail shell, pry it open to reveal the meat, lift out the meat, and discard the black vein down the center. Cut the tail meat into medallions. Crack the claws and extract the meat, leaving the pieces whole for a showy presentation, if desired. Refrigerate the lobster meat until needed.

Transfer the pesto to a bowl and stir in the vinegar and add enough of the 1/4 cup olive oil to make a spoonable vinaigrette. Season to taste with salt and pepper.

To serve, cut the potatoes into 1/4-inch-thick slices or into halves if they are very small.

In a bowl, combine the potatoes, green beans. and about 1/2 cup of the vinaigrette and toss to coat evenly. Distribute the potato mixture evenly among salad plates. Top with the lobster, again dividing evenly, and drizzle with the remaining vinaigrette, Garnish with tiny cherry tomatoes, if desired. Serve immediately.