Put the potatoes into a medium saucepan, for which you have a lid. Add just enough water to cover, salt generously and place on medium-high heat.
Bring to a boil, then decrease the heat to medium-low and cook, covered, for 20–25 minutes, until just tender.
Drain and, once cool enough to handle, remove the skins from the potatoes and slice into ¼-inch/½cm-thick rounds. Set aside in a medium bowl.
Put 2 tablespoons of the oil into a medium sauté pan and place on medium heat. Add the onion and cook for 12–15 minutes, stirring regularly, until caramelized. Add the garlic and stock, bring to a simmer, then remove from the heat.
Add the mustard, vinegar, 1¼ teaspoons of salt, the pepper, and another 2 tablespoons of oil. Whisk to combine, then pour the mixture over the potatoes.
Mix gently but thoroughly: it will look wet (and some of the potatoes will break up), but this is normal. Set aside for about 2 hours, for the potatoes to soak up about half the broth, and then stir in the finely chopped chives.
Meanwhile, wipe clean the sauté pan and place on medium-high heat. Add the pancetta, decrease the heat to medium-low, and cook for 10–12 minutes, stirring occasionally, until the pancetta is crispy. Using a slotted spoon, transfer to a plate lined with paper towels— leave about 1 tablespoon of the fat in the pan— and set aside. Once cool, finely chop the pancetta into crumbles.
Add the remaining 1 tablespoon of oil to the fat in the pan, along with the paprika. Stir for 30 seconds or so, until fragrant, then remove from the heat.
When ready to serve, fold the cucumber into the potato salad and transfer to a serving plate. Scatter the pancetta crumbles over the top, along with the cut chives. Spoon on the paprika oil and serve.