In a large saucepan, place the potatoes, 2 teaspoons salt and enough water to cover by 1 inch. Bring to a simmer and cook until the potatoes are tender, 12 to 15 minutes. Add the green beans and cook for 3 minutes. Drain and let stand.
Meanwhile, in a large bowl, combine the vinegar, lemon juice, sugar, mustard, pepper and 1 teaspoon salt. Whisk in the olive oil until combined.
Toss the potatoes and beans with the dressing to coat. Using a fork, mash several potatoes. Toss in the scallions and thyme. Season with more salt and pepper to taste and top with thyme sprigs. Serve warm or at room temperature.