Place the potatoes in a large pot or Dutch oven. Fill the pot with cold water, so that the potatoes are covered by at least 2 inches of water.
Cover, bring to a boil, then reduce to a medium boil. Cook until the potatoes are tender when pierced with a fork, about 15 minutes, being careful not to overcook.
Drain in a large colander. Place a clean kitchen towel over the colander to allow the potatoes to steam. Set aside.
Heat the same pot or Dutch oven over medium heat. Add the bacon and cook until crisp, flipping once, about 5 minutes.
Remove the bacon and set aside. Pour all of the bacon fat into a heat-proof container. Set aside.
While still warm, slice the potatoes into quarters. Add them to the pot along with any drippings left in the pan. Pour over ¼ cup apple cider vinegar, stirring to combine. Cover and allow to sit for 5 minutes.
Meanwhile, make the dressing by whisking together 2 tablespoons of the still-warm, rendered bacon fat, 2 tablespoons olive oil, honey, brown mustard, whole grain mustard, and salt until well combined. Add ⅓ cup apple cider vinegar, whisking until smooth.
Pour the dressing over the warm potatoes, tossing to coat. Add the chives and toss again. Allow to sit for at least 10 minutes for the flavors to combine and the potatoes to absorb the extra dressing.
Serve warm or refrigerate for up to 5 days. Allow to come to room temperature before serving.