Bring a medium saucepan of water to a boil and salt the water generously. Simmer the potatoes until just tender, 15 to 20 minutes. Remove the potatoes from the pan and let them cool.
In a small bowl, whisk the vinegar with a few big pinches of salt and pepper to taste. Slowly drizzle in the olive oil, whisking constantly. Taste and add more salt if you like.
When the potatoes are cool enough to handle, slice them into 1/2-inch rounds. Arrange the lettuce on a small platter.
Put half the vinaigrette in a medium saucepan and set it over medium heat. Once it is hot, add the chicken and potatoes to the pan. Cook, turning everything over gently with a spatula every couple of minutes, until the chicken is warmed through, and both the chicken and potatoes are crisp around the edges.
Drizzle a few spoonfuls of the remaining vinaigrette over the lettuce and then arrange the chicken and potatoes over the top. Sprinkle the chives over everything and serve immediately, with the remaining vinaigrette on the side.