In a large saucepan, combine the potatoes, thyme sprigs, and 2 tablespoons of the salt. Add enough cold water to cover completely. Bring to a boil over high heat. Reduce the heat to maintain a simmer and cook until the potatoes are just tender enough to be pierced easily with a skewer or the tip of a sharp knife, about 10 minutes.
Drain the potatoes. While they are still hot, peel them with a small, sharp knife, protecting your hand from the heat with a folded kitchen towel. As each potato is peeled, cut it crosswise into 1/4-inch slices, letting the slices fall into a mixing bowl.
Add to the hot potato slices the onion, remaining salt, sugar, pepper, vinegar, and oil. Stir gently but thoroughly with a large spoon to combine the ingredients.
Before serving, heat the broth in a large saute pan over medium-high heat. Add the potato salad and stir it gently just until the potatoes are thoroughly heated. Garnish with parsley or chives and serve immediately.