Place scrubbed whole potatoes in a large pot and cover with cold, salted water. Bring to a boil over high heat, then cook until tender, 15 to 18 minutes; drain and set aside.
Meanwhile, cook bacon in a large skillet over medium heat until crisp, about 9 minutes. Remove bacon to paper towels to drain; reserve the fat in the skillet.
In a small bowl, whisk 2 Tbs. reserved bacon fat (discard the rest), the Dijon mustard, and the vinegar until combined. (The dressing will taste very salty because of the bacon fat, but the flavor will balance once you add the potatoes!) When potatoes are cool enough to handle but still warm, cut them into bite-size chunks and place in a large bowl. Pour dressing over the warm potatoes and gently toss to coat. Sprinkle with minced chives and toss again. Crumble the cooked bacon over the top.