Preheat the oven to 350 degrees F. Arrange the potatoes in a single layer in a baking dish and put in the center of the oven. Bake until the potatoes are tender and can be pierced with the tip of a knife, about 50 minutes.
Meanwhile, combine the parsley and basil in a food processor or blender. Season with salt and white pepper, add 2 tablespoons water and blend. With the motor running, pour the olive oil through the top in a slow, steady stream. Add the sugar and taste for seasoning. Pulse to combine and set aside. (This doesn't need to be a smooth puree: It should be a little rustic and chunky.)
In a large bowl, combine the vinegar, shallots, mustard, capers and brine, and the gherkins and brine. Add half of the parsley-basil puree and toss. Taste for seasoning.
When the potatoes are done, remove them from the oven and let cool for a few minutes. Peel half of the potatoes (some skin adds a nice flavor to the salad). Cut any big potatoes into quarters and cut the rest in half. Put the potatoes in the bowl with the dressing; season lightly with salt and pepper and gently toss. Serve at room temperature with the remaining herb puree on the side.