Drop the white and red potatoes into a large pot of boiling salted water and cook for 20 to 30 minutes, until they are just cooked through. Drain in a colander and place a towel over the potatoes to allow them to steam for 10 more minutes. As soon as you can handle them, cut in 1/2 (quarters if the potatoes are larger) and place in a medium bowl. Toss gently with the wine and chicken stock. Allow the liquids to soak into the warm potatoes before proceeding.
Combine the vinegar, mustard, 1/2 teaspoon salt, and 1/4 teaspoon pepper and slowly whisk in the olive oil to make an emulsion. Add the vinaigrette to the potatoes. Add the scallions, dill, parsley, basil, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper and toss. Serve warm or at room temperature.